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## Research Project Probability Statistics

Research Project probability statistics

Annmarie Persaud

[Institutional Affiliation(s)]

Author Note

Research Project Probability Statistics

In the U.S. healthcare system, community health workers (CHWs) play a vital role in delivering care by acting as a liaison between the community and health services, and by providing resources, information, and options to them regarding their health.

In the study by Ingram et al. (2017), electronic health records (EHRs) are used to investigate the role played by CHWs in primary care focusing particularly on chronic disease-related indicators of health. It is hypothesized that EHRs aid in improving patient management and promote inter-staff communication. For this purpose, EHRs of a sample of 32147 chronic disease patients were examined between the years 2012 and 2015 in which health-indicators were used as variables along with the patient’s level of contact with CHWs, in two groups.

The outcome variables were estimated according to within-group and between-group results through a covariance analysis of the mean of the observations between groups in which one group saw a CHW at least once, and one group did not see any CHW. The findings indicated that within-group there was a statistically significant improvement in health indicators associated with chronic disease after they had contacted CHWs. The within-group analyses showed that HbA1c mm/mol levels decreased to 0.15 with a 95% confidence interval (CI=-0.2, -0.006) and a P-value of 0.001. Whereas cholesterol saw a decrease of 11.9 mg/dl and BMI decreased by 0.22, with a 95% confidence Interval (CI=-13.5, -10.2) with a P value of less than 0.001. However, in the between-group analysis, the only statistically significant difference between non-CHW and CHW groups was in patients' diastolic blood pressures which was 0.38mmHG, with a 95% CI (CI=0.09, 0.67) and a P-value of 0.01.

The results demonstrated that patients saw significant improvements in clinical outcomes in within-group settings after contacting CHWs, while the between-group findings did not show much considerable improvement, except for one outcome, in the outcomes associated with CHW exposure.

References

BIBLIOGRAPHY Ingram, M., Doubleday, K., Bell, M. L., Lohr, A., Murrieta, L., Velasco, M., . . . Carvajal, S. C. (2017). Community Health Worker Impact on Chronic Disease Outcomes Within Primary Care Examined Using Electronic Health Records. American Journal of Public Health, 107(10), 1668-1674.

Subject: Maths

Pages: 1 Words: 300

## Algebra

Algebra

Student’s Name

Institution

Date

Algebra

In mathematics, a solution set is described as the set of value which makes a certain set of equation. It is also regarded as the set of values that satisfy a given inequality. It therefore, means that every value in the solution set can satisfy the inequality. The solution set is also a set of ordered pairs which make equation true. For example 2x+3 greater than 7 and in this case X is regarded as a natural number. Compound inequality contains two inequalities which are joined together by words at the intersection of inequality of every solution. In order to solve compound inequality, it is necessary to separate the inequality into two inequalities and then determine whether it is required to have a union. Therefore, both statements are supposed to be true and the same. The two types of inequality are conjunction problems and disjunction problems, which sometimes appear as two simple inequalities.

Graph: Simple line graph

In the simple line graph in diagram 1 above, there are four intersections where the meeting occurs. However, AND or OR are used to for the separation of two simple inequalities. However, by conducting calculation to solve inequalities, the formula requires that the problem to be solved individually It means that the application of AND and OR are used to solve inequality problems by applying the formula in each case separately. However, intersection is the point where two inequalities meet together by words. It separates the two inequalities and therefore, it is essential for the solving inequalities problems.

References

Tyler L. Wallace (2015). Compound Inequalities

www.wallace.ccfaculty.org › book › 3.2 Compound Inequalities.pdf

Subject: Maths

Pages: 1 Words: 300

## Astronomy

1.In astronomy, light units are a convenient way to measure distances. Light travels at a speed of approximately 3×108 m/ s which means that one light second is approximately 3×108 m. Below we list seven distances in light-units and nine physical sizes. Match each distance to one size.

light hour

light second

light year

light millennium

light minute

light century

light nanosecond

across a small galaxy

across the known Universe

between planets in the outer solar system

across a child's limb

across a medium-sized country

between planets in the inner solar system

between stars in the center of our galaxy

between the Earth and the Moon

2.When astronomers discuss objects that give off light, such as the Sun, we discuss a concept called "luminosity" which is measured in units of energy per unit time. Typically, this is given in "Watts", the same unit of power that is used to rate lightbulbs. The luminosity of the Sun is L==3.8×1026 Watts.

However, the brightness of a luminous object depends on how far away you are from it. This is called the "flux" by astronomers. Flux is calculated using the formula F=L/4πD2.

The flux of light we receive from the Sun here at Earth (a distance of one astronomical unit or =1 AU=1.5×108 km from the Sun) is called the "solar constant".  It's measured to be F=1362 Watts per square meter.

Saturn is 10 AU from the Sun. What is the flux that Saturn receives from the Sun in Watts per square meter? (HINT: There is a very simple way to calculate this using "scaling" techniques and realizing that Saturn is 10 times further from the Sun than the Earth.)

Watts per square meter.

Note: The exactly value of flux we receive from sun on the earth is 1344 watts/m2 but you gave it as 1362 Watts/m2. As per scaling technique the value of flux for Saturn would be 100th time less than value of flux for earth.So I wrote both the answers. One of them is according to the value you provided and the other one I calculated using the formula.

3. Recall that light is a wave and so we can use the relationship that the speed of light is the wavelength of a photon multiplied by the frequency of a photon.  . A famous photon in astronomy is the photon emitted by a hydrogen atom with a frequency of  MHz. What is the wavelength of this photon in centimeters?  centimeters.

The value for the wavelength of above photon is calculated as c/f= λ

3X108 / 1440 X106 = 0.208 m or 20.8 cm

4. Light that is red has a frequency of 4.5 × 1014 Hertz. What is red light's wavelength?

The value for the wavelength red light is calculated as c/f= λ. The value for the velocity was taken equal to the speed of light because red light travels faster than any other color so we took the velocity equal to the maximum speed of light i.e. v=c=3X108 m/s. So, the calculated values for the wavelength of red light having frequency = 4.5 X 1014 Hz.

3X108 / 4.5 X1014 = 6.66X 10-7 meters.

×10

meters

5. Kepler's third law states that for any object in a gravitational orbit,

P2∝a3

where P is the orbital period of the object and a is the average distance between the object and what it is orbiting.

In our Solar System, the natural units are distances measured in astronomical units (A.U.) and orbital periods measured in years. This can be seen for the Earth-Sun system which has an orbital period P=1 year and an average distance  a=1  AU. Using these natural units in the Solar System, the proportionality becomes an equality, so for our Solar System:

(Pyears)2=(aA.U.)3.

Using your mathematical prowess, determine what the orbital period in years would be for an asteroid that was discovered orbiting the Sun with an average distance of 36 astronomical units.

years.

HINT: The following table of so-called perfect squares and cubes can be used to solve this problem without a calculator:

NUMBER

2

3

4

5

6

7

8

9

10

SQUARE

4

9

16

25

36

49

64

81

100

CUBE

8

27

64

125

216

343

512

729

1000

6. We use the equation escape velocity equation that

vescape=11 km/s, formula for escape velocity is vescape= √2G(M⊕)(R⊕)−1.

On this basis, calculate the escape velocity from our Solar System in kilometers per second starting here at Earth. Hint: it helps to express the mass of the Sun as M⊙=3.33×105M⊕ and the distance between the Sun and the Earth as R=1 AU=2.35×104R⊕.  km/s

. Hint: the galaxy has a mass of M=2×1017M⊕ and our Solar System sits at a distance of R=4×1013R⊕  from the center of the Milky Way.  km/s

7. When you step on a scale in the U.S., your weight is typically given in pounds (lb) and you can easily convert pounds to kilograms (kg). Pounds are a measure of force while kilograms are a unit of mass -- you weight depends on the gravity at the surface of the world you are standing on, and the surface gravity depends on the mass and radius of the world. On Earth's surface, the conversion is approximately 1 kg = 2.2 lb.

The surface gravity on the Moon is about 1/6 that of the Earth. If you weigh 220 lb on a scale on Earth's surface, how many pounds would the scale read on the surface of the Moon?

pounds

Note: 36.4 lbf will be the weight on the surface of the moon.

What would be your mass in kilograms on the Earth's surface if you weigh 220 lb on Earth's surface?

kilograms

Note: if the weight on earth is 220lbf then corresponding mass in pounds will be 6.8lb and after applying conversion factor mass in kilograms will be equal to 3.08kg

What would be your mass in kilograms on the Moon's surface if you weigh 220 lb on Earth's surface?

kilograms

Note: Mass will remain same as mass is the base quantity and do not depend upon the value of gravity, so it will remain same on earth and moon. Weight changes on the moon’s surface because it depends upon the value of “g” but mass remains same everywhere.

Take a look at the surface gravity on the various other worlds: http://www.exploratorium.edu/ronh/weight/

If you weigh 220 lb on a scale on Earth's surface, how many pounds would the scale read on Europa?

pounds

Explain why worlds with a very low or very high surface gravity would pose a challenge for humans exploring such worlds.

Subject: Maths

Pages: 5 Words: 1500

## College Algebra

Math 1314 Course Schedule Outline – 15 Week – Spring 2019 Course Week 1 Assignments.

Assignment 1.

Name:

Orientation for Mathlab - Extra Credit

Due:

02/10/19 11:59pm

Last Worked:

02/02/19 1:24am

Current Score:

100% (10 points out of 10)

Question 1 (1/1)

Question 2 (1/1)

Question 3 (1/1)

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Question 8 (1/1)

Question 9 (1/1)

Question 10 (1/1)

Assignment 2.

Name:

1.1 - Lesson 1 Homework

Due:

02/10/19 11:59pm

Last Worked:

02/02/19 3:34am

Current Score:

100% (23 points out of 23)

Question 1 (1/1)

Question 2 (1/1)

Question 3 (1/1)

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Question 19 (1/1)

Question 20 (1/1)

Question 21 (1/1)

Question 22 (1/1)

Question 23 (1/1)

Assignment 3.

Name:

1.2 - Lesson 2 Homework

Due:

02/10/19 11:59pm

Last Worked:

02/02/19 5:06am

Current Score:

100% (20 points out of 20)

Question 1 (1/1)

Question 2 (1/1)

Question 3 (1/1)

Question 4 (1/1)

Question 5 (1/1)

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Subject: Maths

Pages: 18 Words: 5400

## Data Analysis

Epidemiology

[Enter name of Institution here]

Age

Rate (Australia)

Rate (Indonesia)

0-14

1.8989

4

15-39

7.7378

12

40-44

30.8801

38

45-49

64.600

66

50-54

115.1709

120

55-59

188

185

60-64

313

262

65-69

458

436

70-74

656

645

75+

1515

1097

In the above table, we have calculated death rates per 100000 in Australia and Indonesia. We see that rate in Australia is lower than Indonesia in younger age groups whereas in the older age groups, rate is higher in Australia as compared to Indonesia. There is an increase in mortality rates as people get older with the highest rates in people over 75 years of age. There is a considerable difference between populations of two countries. Indonesia has a much larger population but it has lower mortality rates in almost all the age groups.

Country

Total Deaths

Total Population

Total Rate

Australia

43403

23218618

187

Indonesia

215217

244681612

88

There is a much higher death rate in Australia as compared to Indonesia on the aggregate. This is shown in the table above in the last column showing that rate in Australia is 187 per 100000 and that in Indonesia is 88 per 100000.

Graph

Above graph shows a comparison of Death rates in Australia and Indonesia showing that Australian population is much more vulnerable to the disease discussed, especially people over 75 years of age. As far as rates are concerned, highest increase in rate is seen between the last two age groups. For first five to six age groups, there is hardly any difference between the two countries as shown by coinciding lines. In the age group 60-64, there is a slight gap between rates of two countries. In case of Australia, there is a sharp rise in mortality rate in the last two age groups. Although Indonesia has also shown a rise in mortality rates, the hike in its mortality rate is less evident.

Australian population has shown an increase in numbers for first two classes of population. The number of deaths have decreased for the third population group but there has been an increase in rate of deaths because there is a lower population in this age group. From third to fourth age groups, there is a huge increase in number of deceased persons.

Ages

Rates Indonesia

Rates Australia

Population rate Ind

Population rate Aus.

0-14

3.965533086

1.89

10.36626039

4.9406301

15-39

11.77970976

7.74

46.37212322

30.4693614

40-44

37.8675285

30.88

24.94599175

20.3428176

45-49

65.86232207

64.6

39.76701145

39.004834

50-54

120.5091507

115.17

64.69051717

61.8244077

55-59

185.0906939

188

84.1866512

85.50992

60-64

261.9944127

313

97.42786223

116.39531

65-69

436.4063601

458

129.132642

135.5222

70-74

645.3478327

656

142.4934

144.92352

75+

1096.642532

1515

336.0112718

464.196

Above table shows proportion of Australia and Indonesia from World deceased population from cancer using rate of deaths per 100000 as a base. We have divided the rate by 100000 and multiplied it by the population given in world.

There is a sharp increase in individual country rates of death per 100000 people in the age group pf 40-44 years of age. However, there is a slight decrease in the rate of same age group when population standardized rates are calculated which may mean that there are even more people in other countries of the world who suffer from disease in this age group. The highest world standardized rates are experienced by people aged 75+ years for both countries.

Part 2

3614/465280*100000 = 777

1791/156415*100000 = 1145

Rate ratio = 777/1145

= 0.68 which means that people who sat > 8 hours per day had 68% more rate of mortality as compared to those who sit less than 8 hours per day

368 which means that thee are 368 more cases of mortality per 100000 in people sitting more than eight hours than those who sit less than eight hours a day.

(1791/5405)

= 0.331 which means that 33.1 percent of chance is attributable to sitting more than 8 hours per day.

5405/621695*100000

869 which means that there are 869 people per 100000 who die of sitting more than eight hours a day

5405/465280*100000

1162

<8

>8

3614

1791

5405

465280

156415

621659

1791/5405

0.331

g)156415/621659

0.25 which means that population attributable factor is only 25% in case of people sitting more than eight hours per day.

Subject: Maths

Pages: 3 Words: 900

## Descriptive Statistics Project

Descriptive Statistics Project

[Name of the Writer]

[Name of the Institution]

Descriptive Statistics Project

Find the appropriate measure of center. Discuss why the chosen measure is most appropriate. Why did you decide against other possible measures of center?

In this assignment, I choose the values of Departure Delay for further calculations and applying the statistical formulas for analysis. The appropriate measure of center for this set of data is mean. Mean is the most common and widely used method of finding the measure of center. The mean is usually affected by extreme values. If these values are higher than the rest of the data, it could not give correct information. Here we have a normal range will relatively better extreme numbers. The lowest number is -24, and the highest number is 23. The total number of population is 500. When the range of the data is not too higher, we can use the mean to find the center of the measure. It is relatively easy and gives proper information of the average numbers.

Find the appropriate measure of variation. The measure of variation chosen here should match the measure of center chosen in Part 1.

In the variation, we find the difference in the occurrence of data. I used simply the maximum and minimum value of the sample size. The maximum value is 23, and the minimum value is -24.

Find the graph(s) needed to describe the data appropriately. These may be done by hand and inserted into the Word document.

Define a random variable (X) so that your chosen data set represents values of X.

Random variable X is the variable which has some possible outcomes in numerical terms of a random phenomenon. Random variables are of two types known as the discrete and random variables(Shirazi et al., 2016).

Is your chosen random variable discrete or continuous? Explain how you know.

The chosen data set is the discrete random variable because it has no infinite number. All the numbers are countable which are numerical and could be count easily.

Would the Normal or Binomial distribution be a good fit for the underlying sample distribution of X? If one of them is a good fit, state how you would approximate the distribution parameters.

The binomial distribution is quite a different phenomenon from that of the normal distribution. Binomial distribution always has two outcomes such as discrete and continuous. This data is normal because it has many outcomes; a plane could reach in time, early and late.

Calculate the probability that a flight will depart early or on-time.

The probability of a flight that will depart early is 65 percent, and the probability of on-time depart is 7 percent.

Calculate the probability that a flight will arrive late.

The probability of the flight to arrive late is 34 percent.

There are 170 times late arrival of the plane out of the total 500.

Calculate the probability that a flight departs late or arrives early.

The late departs of flights are 27 percent. There are 137 flights which departed late. The early arrival of flights is 58.2 percent.

Assume now that the random variable X=Arrival Time is exactly normally distributed with mean m= -2.5 and standard deviation s= 23. Compute the probability of a flight arriving late based on this new information. Does this contradict your answer from Part 8?

X= 170 the number of flights in time.

M= -2.5

SD= 23

The formula is :

P (X) = x – m/sd

= 172.5/23

P (x)= 7.5

Write a brief description (250–500 words) of the data set including the discussion required in the points above.

The data was regarding the flights timing at the airport. The timings of both departure and arrival of planes. The given data was the record of 2nd-week flights details. The data are given shows that there are a lot of flights which got late in both arrivals and departure. The highest values of the given data of departure are -24 hours which shows the 24 hours early arrival. While the plane was departed late by 23 hours according to the data. The mean of the data of departs flights were -5.269. There were total 500 flights in the week, and the total number of late departs were 137 out of 500. While before in time departure were 36, and early departs were 327 flights. The probability shows us the chances of an outcome. For instance the head and tail in tossing the coin. Random variable X is the variable which has some possible outcomes in numerical terms of a random phenomenon. Random variables are of two types known as the discrete and random variables. A binomial distribution is very different from a normal distribution, and yet if the sample size is large enough, the shapes will be quite similar. The key difference is that a binomial distribution is discrete, not continuous (Tuckwell, 2018). In other words, it is impossible to find a data value between any two data values. Probability measures the chance of a given event occurring. It is a number between 0 and 1 where a probability of 1 is an event that is certain to occur and a probability of 0 represents an event that cannot occur (Probability distributions. (2019).).

References

Probability Distributions. (2019). Erm.ecs.soton.ac.uk. Retrieved 12 March 2019, from http://www.erm.ecs.soton.ac.uk/theme7/probability_distributions.html

Shirazi, M., Lord, D., Dhavala, S. S., & Geedipally, S. R. (2016). A semiparametric negative binomial generalized linear model for modeling over-dispersed count data with a heavy tail: characteristics and applications to crash data. Accident Analysis & Prevention, 91, 10-18.

Tuckwell, H. C. (2018). Elementary applications of probability theory. Routledge.

Subject: Maths

Pages: 3 Words: 900

## Earning And Spending Assigment

﻿Student’s Name:

Professor’s name:

Course Code:

Date:

Earning and spending assignment

This report illustrates the detail of the contract awarded to the company by the pizza company to handle the operation of the telephone. The telephone contract includes the receiving of pizza orders from the clients by phone and forwarding to the Pizza Company for delivery. It requires a long hour work and therefore, several factors must be considered before the deal is accepted. However, to determine the appropriate mode of payment, the calculation was done and the decision was made based on the return and hours. The payment per hour with overtime (wages) is a good mode of payment, which is more viable but it is the best based on the calculation. Pay per piecework gives employee leverage to work and earn more and based on the calculation it is the best mode of payment, which is accepted under the condition and terms of the work.

Pay per piecework is general payment on the number of worked completed or done in a day, In this case, it does depend on the number of pizza ordered by clients or calls receive per day and therefore, it is less risky and protects an employee and gives an individual an opportunity to make more money. It is, therefore, the best payment method, which I will choose as my mode of payment during the working contract with the new company. Pay per piece of work is calculated based on a commission on workload and retainer. And therefore, what is earned depends on the value of products sold in a day. However, pay per hour is calculated based on the hours, and overtime. With pay per piece of work, I would be able to work for several hours including the overtime and make earn a lot as well. It means that when the company’s sales increases, the wages earned by employees increases as well. This is likely to result in a higher pay considering that the calculation of the payment I based on the hour and the value of the workload. The payment based on the hours completed exposes an employee to a lot of risks. It seems that when an employee does not receive any order then it means the employee would no earn. The payment per workload is more beneficial to employees compared to pay per hour because helps a company to reduce wages. But pay workload with retainer would be the ideal method of payment for this contract to be done effectively.

Calculation

Pay per hour with overtime

Week 2:

Wages = hours work X rate per hour + overtime

Week2: [15 *25] + [3*25]

375 + 75

\$ 450

Week 3:

Wages = hours work X rate per hour + overtime

[25*20] + [1.5 *25]

500 +37.5

Week 3 wages = 537.5

Week 4:

Wages = hours work X rate per hour + overtime

[12.5*25] + [6*25]

312.5 + 150

\$ 460.5

The total payment for one month would be

=\$450 + 537.5 +460.5

=\$ 1,448.00

2. Wages paid based on piecework

Week

Total Value of orders

Retainer

Commission % of orders

Total weekly payment

1

\$15,000

115

5%

\$865

2

\$12,000

115

5%

\$715

3

\$19,500

115

5%

\$1,090

4

\$11,700

115

5%

\$700

Table1: Wages paid based on the piece work

Payment for the work done based on hourly pay rate

Week

Standard hours worked

Overtime hours worked

Number of orders taken

Standard hourly rate

Overtime rate

Rate per order taken

Weekly Wage

Weekly piecework

2

15

3

495

25

25

5%

450

7425

3

20

1.5

530

25

25

5%

537.5

10600

4

12.5

6

481

25

25

5%

462.5

6012.5

Table 2: Hourly pay rate and weekly payment

Discussion

Payment per piece of work would be the best payment method for telephone services. Calculation of the payment based on the piecework indicates that it is being paid higher than pay per hour rate with overtime. Based on the calculation, pay per piece work for week 1 is \$865, week 2 is \$715, week 3 is 1090 and week four is \$700. It gives a total of \$3370 per month. The pay per hour with overtime payment are week two \$450, week three is \$537 and week four is \$460.5, with a total of \$ 1,448.00. In this case, it seems payment per piecework earns a lot of wages at the end of every week. And therefore, it is the best method of payment, which would be used as the payment method in this agreement. Payment per piece work is the best since it factors in the production rather than the time an employee spends at work. It means that the work the company receives a day or in a week, the higher the earning or wages an employee will receive at the end of the week. It, therefore, gives an employee an opportunity to work extra hard and earn more wages.

Conclusion

It is, therefore, my pleasure to inform you that I have accepted the job offer at your company. It would be of my interest to work under the payment term of pay per piecework payable after every seven days or weekly. The payment terms, which include a commission of 5% of the total value of orders made in a week, with a retainer of \$115 per week. The payment should factor in the workload and other benefits, which might be offered. With these terms of payment, I would be willing to start the work anytime.

Subject: Maths

Pages: 3 Words: 900

[Author Name(s), First M. Last, Omit Titles and Degrees]

[Institutional Affiliation(s)]

Author Note

Mathematical Investigation and Analysis (1)

Here ‘a’ is negative because the curve of the bridge is in the downward direction. The value of ‘c’ is 0 to maintain the structure of the bridge. Actually, point c represents the point of intersection between the curve and y-axis, i.e. between the road and the bridge. So it has to be zero in this case.

‘a’ is less than 1 because we want to draw the parabola in downward direction according to the requirement of bridge. As stated before, if its value is greater than 1 then the parabola will open in the upward direction, hence will disturb the construction of the bridge. After finding the equation of parabola by using either method, we can determine its roots by using the quadratic formula. The vertex is the point of the parabola, which is on maximum height, and it is given in the question. The axis of symmetry can be calculated by the value of 'x' in the vertex.

Mathematical Investigation and Analysis (2)

In order to find the equation of parabola using the trial and error method, we need multiple points on it. For the calculation of this point, we can use different techniques, including different kinds of interpolation. By doing all the maths, the final equation of the parabola is given by

y = -0.001x2 + 2.7x + 0

This equation is the true depiction of the required bridge and is verified by using Desmos.

Mathematical Investigation and Analysis (3)

Once we have calculated the equation of the parabola, we can find out other points of intersection between the line y=50 and the parabola by simultaneously solving the two equations. The two points of intersection are (20, 50) and (250, 50). Now to calculate the distance between the consecutive cables, we will divide the distance by 19. The distance between the consecutive cables is 12.105. Now, in order to calculate the coordinates of the cable with road, we will add 12.105 in all x- coordinates of the cables. Next, to find the coordinates of the cable with arch, we put the values of x-coordinates of all cables in parabolic equation. The length of the cable can be calculated in the following two ways.

By subtracting the y-coordinates found in the previous step by 50.

By using the distance formula in which the two points will be the points of intersection between the cables, roads and between the cables and the arc.

Mathematical Investigation and Analysis (4)

The reason the algebraic method is preferred over the trial and error method, in this case, is that in the former method, we need only two points in the parabola to determine its equation. However, the trial and error method demands multiple points to find the equation. To calculate the parabolic equation using algebraic, we use the following two points and the general equation.

y = a(x-h)2 + k

(135, 182.25) and (270,0)

Putting these two points in the equation, we will get

a = - 0.01

Hence the final equation of the parabola is

y= - 0.01 (x-135)2 + 182.25

Conclusion

The similarity between the two results of both methods indicates that our analysis for the construction of bridge is true. Also, the Desmos graph approves our results.

Subject: Maths

Pages: 2 Words: 600

[Author Name(s), First M. Last, Omit Titles and Degrees]

[Institutional Affiliation(s)]

Author Note

Introduction

In this task, we are required to analyze a bridge and its parameters. Different cables are used in the construction of this bridge on the road. We need to calculate different parameters of these cables, along with the parabolic equation of the bridge. For determining the equation of bridge, we have to use two methods i.e. ‘trial and error method’ and the simple ‘algebraic method’. In addition, we have to do some simple reasoning regarding the different signs of the parameters. All the details are explained below under their specified headings.

Mathematical Investigation and Analysis (1)

The general equation of a parabola is, ‘y = ax2 + bx + c’. Now, in this formula, if ‘a’ is positive, then it means that parabola opens in an upward direction, and if it’s negative, then parabola opens in a downward direction. In this case, the shape of the bridge can be represented by the parabolic equation: ‘y = ax2 + bx + c’ Now, as for the bridge, the face of the parabola is downwards i.e. it opens in the downward direction. Hence, its ‘a’ value is negative. Also, the value of ‘c’ is ‘0’ because the bridge intersects with the y-axis at the origin. If ‘c’ had any value other than zero, then the bridge would have moved upwards or downwards, making it unable to stand. The value of ‘a’ is less than ‘1’ because if it’s greater than ‘1’, then the curve would open in the upward direction and the desired structure of the bridge could not be achieved. If its value is equal to zero, then the curve would become flat and its shape would be distorted. Hence, its value will always be less than ‘1’, in fact in this case, as the curve is downwards, its value will be less than ‘0’ i.e. negative. Once, we know the equation of the parabola, we can find its roots simply by using the quadratic equation. i.e. ‘x = - b± b2-4ac 2a’. For finding the vertex, we will first take the derivative of the equation of the parabola and then put it equal to zero to solve it for the value of ‘x’. We will then put this value of ‘x’ in the original equation of parabola to get the corresponding value of ‘y’. These two values of ‘x’ and ‘y’ will be our required vertex. i.e. (x, y). The value of ‘x’ calculated for finding the vertex will decide the axis of symmetry for the given parabola.

Mathematical Investigation and Analysis (2)

To find the equation of the parabola using the ‘trial and error method’, we first have to calculate at least 5 points of the parabola. This can be done by using quadratic interpolation. After doing the interpolation, we will find the common differences between the ‘y-values’. And then, by equating those differences to the general form of quadratic equation, we can determine the values of ‘a’ and ‘b’. By doing the analysis, as stated above, the values of a & b, we got are,

a = -0.0099 and b = 2.7000111.

Substituting these values in the general form of quadratic equation, our final quadratic equation looks like,

y = -0.0099x2 + 2.7000111x + 0 (‘c’ is replaced by ‘0’)

For the verification of our equation, we plotted it on Desmos. The result we got on Desmos is attached below,

Mathematical Investigation and Analysis (3)

Mathematical Investigation and Analysis (4)

The ‘trial and error method’ is not suitable here, as it is very lengthy because we have to find multiple points on the parabola first. On the other hand, by using the vertex and a point on the parabola, we can determine the equation of parabola quite easily. The derivation of the formula is shown below:

Conclusion

After the verification of our formula by both the methods and by Desmos, we come to this conclusion that our analysis is true. All the parameters of cables and bridges, calculated above support our result.

Subject: Maths

Pages: 2 Words: 600

## How Will I Use Math In My Chosen Profession?

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My professional major is in economics and statistics. Mathematics has always been one of the major elements of study in economics since olden times. When goods valued for goods during the trade. Though the introduction of money as the acceptable business legal stamp all the traded goods acquired a numerical value. Doing mathematics calculations became even more important. The use of mathematics has become more important in the competitive markets today where every chance are created to earn a business profitable net income. Statistics in the economical world has been frequently used to find market analysis (Levi et al 56).

Mathematical economics is usually used in statically commercial endeavors to enter and pull concern operations. Several commercial organizations use mathematics in accounting, stock checking, calculation of the revenues and fiscal analysis. The kind of mathematics usually used in commercialism is mainly simple arithmetical calculations, algebraic expressions, statistical equations and chance. The commercial market can be made to be more productive but even using advanced mathematics like the matrix, commercial mathematics and additive scheduling.

Consumer mathematics is usually of great significance since it includes a group of topics that are on a practical basis. These topics are used in cooperative settings and mundane life. These practical applications involve the use of previously stored data including the commercial value in the price reduction in the past, markups and markdowns, loans on properties, pay sheet calculations and concern recognition. To interoperate or predict the future direction that may be taken by the business.

The importance of mathematics in economics and business is that mathematic is usually used in the proceeds of our day to day life. Many marketable professional occupations such as company manager, statistical analyst, economics analyst and concerns advisers, have to have a good back group in basic mathematics and even advanced computations that involve mathematics.

The four basic mathematical signs that always used in the daily calculations of business transactions, income statements and profits and losses. These signs include the addition subtraction, division and multiplication. Analyzing the market is one of the basic components of concerns. This is done by observing the wants of the consumers, accessing the competitors strengths and weaknesses in the market. This information may be got through the use of questioners to gather all the relevant information that you may need. This may be convenient by the use of the census bureau and use the statistical knowledge to change the information into per centum mode and finally come up with the customers tendency and place it off a market graph. This, therefore, helps measure the economic performance and all this is usually acquired by having enough information on mathematics (David et al 45).

In the calculation of the gross domestic product which is referred to as the income in a country added to the revenues collected pus the total investment within the country added to the difference of the exports and the imports. You will always need mathematics to calculate all these summations. Mathematics will also help an encomiast calculate the GDP per capita. It will also help to broadly understand the inflation and interest rates in and economy with the use of statistical knowledge (Levi et al 32).

The knowledge on basic maths will help make it easier for proper bookkeeping of the revenue enhancement records which is a basic necessity for businesses and corporations. The revenue enhancement includes collective revenues like the unemployment revenue enhancement, the gross revenue enhancement and the security revenue enhancement which must always be monitored from time to time. And when one does not have the basic mathematical knowledge it becomes easier to make errors during the calculations of these revenue enhancements. Therefore when the evaluation is done on the records unrealistic profits of losses may be found in the reports.

Algebra is usually the kind of maths that are used on the concentration and statistics. Algebra may be used in the calculations of the entire cost and the gross. Advanced calculus is also used to derive complex functions and to design the public service corporation curves (Richardson et al 56). Having statistical knowledge enables the economics experts to do the estimation and realize the chance of an economic event happening. Therefore mathematical knowledge can be used to help make serious decisions in an organist ion.

Economists can find the possibility of an event happening. In the instance of the institute of health wanted to know the reason for an increase in the maternal mortality rate and the frequent use of the C- section as a means of giving birth. The statistical experts determined that the female patients could carry overweight babies and the lack of proper maternity clinical care. Based research that was done by the economist it was realized that the women took overweight drugs that as a result caused complications to the children.

Economists are trained mathematically to handle calculations with imperfect information. Mathematics can also help economics experts to tell irrational human behavior (David 98). Therefore mathematics is an open ration system that has everything worked out through it. When one is well versed with mathematical knowledge, he will no probably suffer from the deficiency of jobs in the commercial market. There are also new findings monthly on mathematical concepts. Mathematics is largely accepted to be in a relationship with all the scientific disciplines.

In conclusion, mathematics helps the learner acquire broad skills in communication, analysis, research, logical thinking, problem-solving, interpretation and finally decision making (David et al 51). It is also an important unit in the current job market since most employers give priority to graduates with mathematical qualifications because of their resourcefulness (Levi et al np).

References

Franke, M. L., Carpenter, T. P., Levi, L., Fennema, E. (2001). Capturing teachers generative change A follow-up study of professional development in mathematics.American educational research journal,38(3), 653-689.

Borko, H., Jacobs, J., Eiteljorg, E., Pittman, M. E. (2008). Video as a tool for fostering productive discussions in mathematics professional development.Teaching and teacher education,24(2), 417-436.

Clarke, David, and Hilary Hollingsworth. Elaborating a model of teacher professional growth.Teaching and teacher education18.8 (2002) 947-967.

Darling-Hammond, L., Richardson, N. (2009). Research review/teacher learning What matters.Educational leadership,66(5), 46-53.

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Subject: Maths

Pages: 3 Words: 900

## Linear Programming Case Study

Week 8 Question 1:

The problem under consideration has a total of five constraints. The data provided shows that the problem is a maximization problem in which the profit generated by selling the beer need to be maximized.

There are three variables that need to be adjusted for this maximization problem. The number of beers for each type constitute the variables. The constraints for the problem include the budget, the storage capacity and the maximum customer demand provided in the question. The objective function in this problem is defined as the cost function depending upon the prices of each type of beer being sold. The solution as found in Microsoft Excel is shown below. The excel file is attached as well.

The number of beers that need to be sold for maximum profit can be seen in front of the variables x1, x2 and x3. The maximized profit can be seen in front of the cell stating objective function. The way that the constraints have been met can also be seen from the given excel sheet.

Week 9 Question 5:

The office manager for the Gotham Life Insurance Company orders letterhead stationery from an office products firm in boxes of 500 sheets. The company uses 6,500 boxes per year (250 working years). The annual carrying cost is 20% of the price of a box of stationery and ordering cost is \$28. The following discount price is provided by the office supply company.

Order quantity: 1-499 Price per box: \$16

Order quantity: 500-999 Price per box: \$14

Order quantity: 1000-1499 Price per box: \$13

Order quantity: 1500 or more Price per box: \$12

A. What is the optimal order quantity?

B. How many orders are placed per year?

C. What is the cycle time?

D. What is the reorder point if the lead time is two weeks?

Economic Order Quantity (EOQ)

If using the economic order quantity for placing orders, the company has to monitor its inventory continuously. Such a policy minimizes the total inventory-related costs. These costs include the cost of purchasing, ordering and holding inventory.

A. Finding the optimal policy

We will calculate the EOQ for each price as

Q=√2∗Annual Demand ∗ Ordering Cost/Carrying Cost ∗ Purchase Price

If a particular quantity is not feasible for the price it is calculated for, we will adjust it to the nearest feasible quantity. Then, we will calculate the total cost of each option as

C(Q)=Annual Demand ∗ Purchase Price + Annual Demand / Q ∗ Ordering Cost + Q/2∗ Carrying Cost ∗ Purchase Price

If the price is \$16 per box, then

Q=√2∗6,500∗\$280.2∗\$16=337 boxes

The total cost is

C(337)=6,500∗\$16+6,500337∗\$28+3372∗0.2∗\$16=\$104,000+\$540+\$540=\$105,079C(337)=6,500∗\$16+6,500337∗\$28+3372∗0.2∗\$16=\$104,000+\$540+\$540=\$105,079

If the price is \$14 per box, then

Q=√2∗6,500∗\$280.2∗\$14=361 boxes

This order quantity is not high enough to get this price, and so we adjust it to 500 boxes.

The total cost is

C(500)=6,500∗\$14+6,500/500∗\$28+500/2∗0.2∗\$14=\$91,000+\$364+\$700=\$92,064

If the price is \$13 per box, then

Q=√2∗6,500∗\$280.2∗\$13=374 boxes

This order quantity is not high enough to get this price, and so we adjust it to 1,000 boxes.

The total cost is

C(1,000)=6,500∗\$13+6,5001,000∗\$28+1,0002∗0.2∗\$13=\$84,500+\$182+\$1,300=\$85,982

If the price is \$12 per box, then

Q=√2∗6,500∗\$280.2∗\$12=389 boxes

This order quantity is not high enough to get this price, and so we adjust it to 1,500 boxes.

The total cost is

C(1,500)=6,500∗\$12+6,5001,500∗\$28+1,5002∗0.2∗\$12=\$78,000+\$121+\$1,800=\$79,921

The lowest cost is achieved at the order quantity of 1,500 boxes. Hence, the optimal order quantity is 1,500 boxes

B. Finding the number of orders per year

Average Number of Orders Placed per Year=Annual Demand/Q=6,5001,500=4.33 orders per year

C. Finding the cycle time

Cycle Time = Number of Days in a Year/Average Number of Orders Placed per Year=250/4.33=58 days

D. Finding the reorder point

Reorder Point=Lead Time in Working Days ∗ Daily Demand=10∗6,500/250=260 boxes

Question 3:

Assume the carpet store makes N orders per year

Then the quantity of carpet ordered will be 10000/N yards.

Since no particular details about carpet inventory are given, you will assume that the inventory is

completely depleted during each order period, just as a new order is received. This means the average

inventory during the order period is 1/2 of the order quantity, or 5000/N yards.

The total annual cost (C) of ordering and warehousing the carpet is...

(at \$8 per yard)

Plotting this function shows that a minimum cost is achieved somewhere around 5 orders per year. Find the exact value by setting the derivative of Eqn 1 equal to zero, and solve for N.

The negative solution for N is meaningless in this problem, so N = 5 is the number of orders that will (5) (6) (8) (9) (7) (4) result in the lowest overall cost for the year when the carpet costs \$8/ yard The inventory cost (I), at N = 5 orders per year is...

When the carpet cost is \$6.50/yd, the cost equation changes, but the optimum number of orders per year does not. I'll let you recalculate eqn 1 based on a carpet price of \$6.50 and prove it to yourself. The inventory cost will change however, since you will only need 2 orders of 5000 yards to satisfy the annual demand. So the inventory cost at N = 2 is,

Some folks would add the average cost of the carpet in inventory to this value, but your problem statement gives no indication of whether or not to do this. Do not let the inventory cost be the only consideration for what number of orders to make. Go back and use the first equation to determine annual cost at \$8/yd and 5 orders per year...

at \$6.50/yard and 2 orders per year.

Week 10 question 3

Week 10 question 2

Subject: Maths

Pages: 12 Words: 3600

Setting up a Budget for Personal / Family Needs

Setting up a budget can be found beneficial cover up personal and family needs. It is a fact that setting up a budget for the personal and family needs can be a complex task and duty when it comes to giving time to it to prepare or set and updating it accordingly. But it ensure to provide a large number benefits to the person and family. It benefits because its benefits both the personal and family by assisting them to manage and cover up their financial needs and have the financial issues catered. Furthermore, setting up a budget allows the person and family to have a proper spending plan at hand for the money one has to spend. It makes sure that the person would have enough money almost all the time for the purpose spend for the things he/she needs and wants to get. It also helps the person to have the required amount of money for spending on the important things that are related to one’s personal needs of needs of the family (Fairchild, Fairchild, & Jones, 2019). Apart from this, setting up a budget allows one to stay away or out of debt and enable to work the way out of the debt or liabilities that one already have.

There are many reasons due to that it becomes clear that setting up a budget for personal as well as family needs is highly crucial. Hence, some of the main reasons that clarify this statement are as follows.

Setting up a budget enables the person to make sure that the money is getting spent on the things on which it has to be. In other words, it assists assurance that you do not spend money on things you do not have to spend on.

In terms of family, setting up a budget is crucial because it allows and teaches members of the family the worth and importance of money for some particular family needs. Apart from this, it also helps the family and members of the family to identify the areas where they have to spend to cover up particular family needs.

Furthermore, setting up a budget for family and personal enables and assists you to take necessary steps for the purpose to curtail the expenses on the things or items that have no much important and even unnecessary up to an extent (Wagoner, 2012).

In short, setting up a budget allows you and your family to have spending plan that further ensures that you and your family have enough financial resources for different needs that are important and required to be covered up (Fairchild, Fairchild, & Jones, 2019).

Issues with Trying To Live Within a Set Budget

It is good almost all the time to set a budget for life and living but one living with a set budget face several issues as well. These issues come to face due to different reasons behind. Some of the main issues that the person trying to live within a set budget have are below.

A possibility exists almost all the time that the person trying with a set budget may face the unexpected expenditures any time in life.

There is a scarcity of extra money that is required for some extra but important activities or needed to be spent in case of urgency.

The person living with a set budget have different personal issues such as the challenge of saying No to others in some cases.

The consistency in the spending every month, week, and day is not possible hence the expenses may vary at times which is also one of the issues for the person trying to live with a set budget (Haegele, 2019).

References

Fairchild, G. B., Fairchild, T., & Jones, L. I. (2019). Personal Budgeting Overview: One Last Pedicure. Darden Case No. UVA-F-1865.

Wagoner, J. (2012). Personal Budgeting: What Are We Trying to Do? Journal of Financial Service Professionals, 66(1).

Subject: Maths

Pages: 2 Words: 600

## MAT 300: Descriptive Statistics Assignment

MAT 300: Descriptive Statistics Assignment

Student’s Name

Institution

Date

Introduction

The article “Suicide rate up 33% in less than 20 years” by Anne Godlasky and Alia E. Dastagir was published in USA Today. It illustrates the increase of suicide among Americans despite the high campaigns and funding. It illustrates that since 1999, the suicide rate in the United States has increased by 33%, which is a worrying trend. It also suggests that more than twice Americans are likely to take their own lives. And the risk factors, which have contributed to increasing suicide, are alcohol and substance abuse, which is rampant across many states CITATION God18 \l 1033 (Godlasky & Dastagir, 2018). The article also pointed out that increase suicide is also directly linked to teens having smartphones. It is also established that suicide is the third leading cause of death in the United States and therefore, it has been regarded as public emergency health.

A descriptive statistic is a brief descriptive coefficient of a given data, which represent the entire data. It is broken down into tendency, which includes mode, median, and means of a given data of a larger sample CITATION Wil18 \l 1033 (Kenton, 2018). The article is a descriptive statistic because the entire data is presented by a section of the data to reflect the happenings in society. The variables were also compared based on the mean and how they relate to the research question. In the article, the suicide mean or rate was directly directed to alcohol and substance abuse, usage of smartphone among teenagers and economic recession. The comparison was done based on their means to determine how suicide relates to the four factors. In order to determine the relationship between suicide and alcohol abuse, the variance of the subject was used as well. It is, therefore, evident that the article was completed using descriptive statistic.

The article can be used by the government for economic and social development planning. It addresses issues which affect the majority of young Americans. And the fact that suicide has been an issue, it can be useful for economic and social planning. The government can determine how to allocate funds in order to address the problem, which affects people. It can also provide the information needed to address alcohol and substance abuse in the country. The article can be used to give a projection of the population, which is used for economic planning of a country. In my current job, it can be applied in establishing community centers to solve the social-economic problem in society. The company can, therefore, use the report to allocate the resources. Most companies which are involved in campaigns to address the problems related to alcohol abuse and suicide usually utilize data or research to address the related problems. The fact that the research analyses the impact of suicide and the trend in society makes the research more useful to several organizations for current and future application. In the future, it can be utilized to structure how a company operates especially companies which deal with alcohol and related products.

The various types of data were used in the research to obtain accurate research findings. Findings are a critical component of the research for accurate information to be obtained various types of data should be used. It widens the scope of the study by providing information, which can reflect the perception of the majority of people in society. The use of limited data is not encouraged because the scope will be limited and the probability of not representing the general population is high. Therefore, the use of various types of data is encouraged because of the accuracy of the findings it helps to obtain.

References

BIBLIOGRAPHY Godlasky, A., & Dastagir, A. E. (2018). Suicide rate up 33% in less than 20 years, yet funding lags behind other top killers. https://www.usatoday.com/in-depth/news/investigations/surviving-suicide/2018/11/28/suicide-prevention-suicidal-thoughts-research-funding/971336002/, 2-35.

Kenton, W. (2018). Descriptive Statistics. https://www.investopedia.com/terms/d/descriptive_statistics.asp, 2-35.

Subject: Maths

Pages: 2 Words: 600

## Math

Math 1314 Course Schedule Outline Assignments.

Name:

3.5 - Lesson 15

Due:

04/07/19 11:59pm

Last Worked:

04/06/19 4:29pm

Current Score:

100% (13 points out of 13)

Questions: 13 Scored: 13 Correct: 13 PartialCredit: 0 Incorrect: 0

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Name:

4.1 - Lesson 16

Due:

04/07/19 11:59pm

Last Worked:

04/06/19 4:54pm

Current Score:

100% (11 points out of 11)

Questions: 11 Scored: 11 Correct: 11 Partial Credit: 0 Incorrect: 0

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Name:

4.2 - Lesson 17

Due:

04/07/19 11:59pm

Last Worked:

04/06/19 5:42pm

Current Score:

88.89% (8 points out of 9)

Questions: 9 Scored: 9 Correct: 8 Partial Credit: 0 Incorrect: 1

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Name:

4.3 - Lesson 18

Due:

04/07/19 11:59pm

Last Worked:

04/06/19 6:39pm

Current Score:

100% (12 points out of 12)

Questions: 12 Scored: 12 Correct: 12 Partial Credit: 0 Incorrect: 0

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Subject: Maths

Pages: 13 Words: 3900

## Math

[Name of the Writer]

[Name of Instructor]

[Subject]

[Date]

Math14-TanB= 34 and 0<B<90Sin12B=±1-cosB2=?We don't have value of cosB so first we have to find cosBAs we know TanB= PerpendicularBase=34, and CosB=BaseHypotenuse So we have to find out Hypotenuse to find value of CosBHypotenuse=Perpendicular2+base2=32+42=9+16= 25=5CosB=45We know that tangent is positive in first quadrant so we will use Sin12B=1-cosB2So Sin12B=1-452=5-452=15×2=110A

15-As we know sin60°=32 , And sinθ is positive because it lies in the first quadrant

So sin12120=1-cos1202=1-(-12)2=2+122=32×2=34=32

16-As we know cos225°=-22 , And cosθ is negative because it lies in the third quadrant

So cos12450=-1+cos4502=-1-02=-12=-22

17-As we know tan135°=-1 , And tanθ is negative because it lies in the second quadrant

So tan12270=-1-cos2701+cos270=-1-01+0=-11=-1

18-As we know cosA=725 , And 0 < A < 90 so cosA will be positive

sin12A=1-cosA2=1-7252=25-7252 =182 ×25=1850=3252 = 35

cos12A=1+cosA2=1+7252 =25+7252=322 ×25=3250=4252 = 45

tan12A=1-cosA1+cosA=1-7251+725=25-72525+725=1832=3242=34

19-As we know cosB=59 , And 0 < B < 90 so cosB will be positive

sin12B=1-cosB2=1-592=9-592 =42 ×9=232

cos12B=1+cosB2=1+592=9+592 =142 ×9=1432

tan12B=1-cosB1+cosB=1-591+59=9-599+59=414=214

20-As we know cosθ=78 , And 0 < θ < 90 so cosθ will be positive

sinθ2=1-cosθ2=1-782=8-782 =12 ×8=123

cosθ2=1+cosθ2=1+782=8+782 =152 ×8=1523

tanθ2=1-cosθ1+cosθ=1-781+78=8-788+78=115=115

21-As we know sinA=0.6=610, And 0 < A < 90 so sinθ will be positive

However first we have to find out cosA to find the half angle identities

sinA = PerpendicularHypotenuse=610, we have to find the Base using the formula H2 = P2 + B2

Base = B = H2-P2=102-62=100-36=64=8

cosA = BaseHypotenuse=810

sin12A=1-cosA2=1-8102=10-8102 =22 ×10 = 225

cos12A=1+cosA2=1+8102 =10+8102=182 ×10 = 3225

tan12A=1-cosA1+cosA=1-8101+810=10-81010+810=218=232=13

22- (4) sin12x= 12sinx is not an identity

23-2 sin40°

24-4 tan250°

25- (1) 1-cosθ

26- (3) -79

27- (2) 2-√3

Subject: Maths

Pages: 2 Words: 600

## Math

Name:

2.1 - Lesson 7 Homework

Due:

03/10/19 11:59pm

Last Worked:

03/05/19 11:17pm

Current Score:

100% (17 points out of 17)

Math 1314 Course Schedule Outline Assignments.

Questions: 17 Scored: 17 Correct: 17 Partial Credit: 0 Incorrect: 0

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Name:

Exam 3 Review - Extra Credit

Due:

04/07/19 11:59pm

Last Worked:

03/06/19 3:51am

Current Score:

99.55% (54.75 points out of 55)

Questions: 55 Scored: 55 Correct: 54 Partial Credit: 1 Incorrect: 0

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Subject: Maths

Pages: 20 Words: 6000

## Math

[Name of the Writer]

[Name of Instructor]

[Subject]

[Date]

Math15- cos2θsinθ+sinθ=cscθ-sinθ

LHS: Left Hand Side

RHS: Right Hand Side

LHS= cos2θsinθ+sinθcos2θ=cos2θ-sin2θ

=cos2θ-sin2θsinθ+sinθLCD: sinθ sinθ

= cos2θ-sin2θ+sin2θsinθ where-sin2θ+sin2θ=0

cos2θ = 1 - sin2θ

=1-sin2θsinθ

=1sinθ- sin2θsinθ

cscθ = 1/ sinθ

= cscθ – sinθ = RHS

Hence proved LHS = RHS

16- 2tanθ-sin2θ2sin2θ=tanθ

LHS=2tanθ-sin2θ2sin2θ

sin2θ=(1-cos2θ)

=2sinθcosθ-2sinθcosθ2(1-cos2θ)

=2(sinθ-sinθcos2θcosθ) 2(1-cos2θ)

=sinθ1-cos2θcosθ(1-cos2θ)

= tanθ = RHS

Hence proved LHS = RHS

17- 2cosθ - cos2θcosθ=secθLHS= 2cosθ - cos2θcosθ

Cos2θ = Cos2θ – sin2θ LCD: cosθcosθ

= 2cos2θ-(cos2θ-sin2θ)cosθ

= 2cos2θ-cos2θ+sin2θcosθ

= cos2θ+sin2θcosθ

Cos2θ + sin2θ = 1

= 1cosθ

secθ=1cosθ

= secθ = RHS

Hence proved LHS = RHS

18- cos2θ+ cosθ+1sin2θ+sinθ=cotθ

LHS=cos2θ+ cosθ+1sin2θ+sinθ

Cos2θ = Cos2θ – sin2θ and sin2θ =2sinθcosθ= cos2θ-sin2θ+ cosθ+12sinθcosθ+sinθ

sin2θ=1-cos2θ

=cos2θ-(1-cos2θ)+ cosθ+12sinθcosθ+sinθ

=cos2θ-1+cos2θ+ cosθ+12sinθcosθ+sinθ

=2cos2θ+ cosθ2sinθ(1+cosθ)

=2cosθ(cos+ 1)2sinθ(cosθ+1)

=cosθsinθ

=cotθ = RHS

Hence proved LHS = RHS

19- sinθ+sin2θsecθ+2=sinθcosθ

LHS=sinθ+sin2θsecθ+2

Sin2θ = 2sinθcosθsecθ=1cosθ

=sinθ+2sinθcosθ1cosθ+2

=sinθ+2sinθcosθ1+2cosθcosθ, LCD=1cosθ/cosθ

=sinθ(1+2cosθ)1+2cosθcosθ

=sinθ1cosθ

=sinθcosθ = RHS

Hence proved LHS=RHS

20- 1+cos2θ1-cos2θ=cot2θ

LHS= 1+cos2θ1-cos2θ

Cos2θ = Cos2θ – sin2θ

= 1+cos2θ –sin2θ1-(cos2θ –sin2θ)

= 1+cos2θ –sin2θ1-cos2θ+sin2θ

cos2θ = 1 - sin2θ

sin2θ = 1 - cos2θ

= 1+cos2θ –(1- cos2θ)1-(1-sin2θ)+sin2θ

= 2cos2θ 2sin2θ

=cot2θ = RHS

Hence proved LHS = RHS

21- sinπ6-θ+sinπ6+θ=cosθ

LHS=sinπ6-θ+sinπ6+θ

sin ( a + b) = sin(a)cos(b) + cos(a)sin(b)

sin ( a - b) = sin(a)cos(b) - cos(a)sin(b)

=sinπ6cosθ-cosπ6sinθ+sinπ6cosθ+cosπ6sinθ

=2sinπ6cosθ

sinπ6= 12

=22cosθ

= cosθ = RHS

Hence proved LHS = RHS

22- cosπ4-θ-cosπ4+θ=√2sinθ

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

LHS= cosπ4-θ-cosπ4+θ

=cosπ4cosθ+sinπ4sinθ -(cosπ4cosθ-sinπ4sinθ)

=cosπ4cosθ+sinπ4sinθ -cosπ4cosθ+sinπ4sinθ=2sinπ4sinθ

sinπ4=√22=2√22sinθ = √2sinθ = RHS

Hence proved LHS = RHS

23- sinπ2+θ-sinπ2-θ=0

LHS= sinπ2+θ-sinπ2-θ

sin ( a + b) = sin(a)cos(b) + cos(a)sin(b)

sin ( a - b) = sin(a)cos(b) - cos(a)sin(b)

=sinπ2cosθ+cosπ2sinθ-(sinπ2cosθ-cosπ2sinθ)

=sinπ2cosθ+cosπ2sinθ-sinπ2cosθ+cosπ2sinθ

=cosπ2sinθ

cosπ2=0

= 0 = RHS

Hence proved LHS = RHS

24- sinπ4+θ+cosπ4+θ=√2cosθ

LHS= sinπ4+θ+cosπ4+θ

Sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

=sinπ4cosθ+cosπ4sinθ+ cosπ4cosθ-sinπ4sinθ=√22cosθ+ √22sinθ+√22cos- 22sinθ

=2√22cosθ

=2cosθ = RHS

Hence proved LHS = RHS

25- tanπ4+θtan3π4-θ=-1

LHS= tanπ4+θtan3π4-θ

tana+b=tana+tan⁡(b)1- tanatan⁡(b)

tana-b=tana-tan⁡(b)1+ tanatan⁡(b)

=tanπ4 +tanθ1-tanπ4tanθtan3π4-tanθ1+tan3π4tanθ

tanπ4=1 & tan3π4= -1

=1 +tanθ1-tanθ-1(1+tanθ)1-tanθ

= -1 = RHS

Hence proved LHS = RHS

26- sinπ4+θcosπ4-θ=1

Sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

LHS= sinπ4+θcosπ4-θ

=sinπ4cosθ+cosπ4sinθcosπ4cosθ+sinπ4sinθ

=√22cosθ+√22sinθ√22cosθ+√22sinθ

= 1 = RHS

Hence proved LHS = RHS

27-a) sin2A1+cos2A=tanA

LHS=sin2A1+cos2A

Sin2A= 2sinAcosA

Cos2A= cos2A – sin2A

=2sinAcosA1+cos2A-sin2A

Sin2A = 1 – cos2A

=2sinAcosA1+cos2A-(1-cos2A)

=2sinAcosA2cos2A

=2sinA2cosA

= tanA = RHS

Hence proved LHS = RHS

b) sin212θ1+cos212θ=tan12θ

LHS=sin212θ1+cos212θ

Sin212θ= 2sin12θcos12θ

Cos212θ= cos212θ – sin212θ

=2sin12θcos12θ1+cos212θ-sin212θ

Sin212θ = 1 – cos212θ

=2sin12θcos12θ1+cos212θ-(1-cos212θ)

=2sin12θcos12θ2cos212θ

=2sin12θ2cos12θ

= tan12θ = RHS

Hence proved LHS = RHS

c) ±1-cos2θ1+cos2θ=sinθ1+cos2θ

LHS=±1+cos2θ1+cos2θ

LCD: 1+cos2θ1+cos2θ

=±1-cos2θ1+cos2θ ×1+cos2θ1+cos2θ

=± 12-cos2θ21+cos2θ2

= sinθ21+cos2θ

=sinθ1+cos2θ=RHS

Hence proved LHS = RHS

Subject: Maths

Pages: 8 Words: 2400

## Math

Math

[Name of the Writer]

[Name of the Institution]

Math

Introduction

The 18th-century economist Thomas Robert Malthus predicted in his book An Essay on the Principle of Population (1798) that London would run out of food as it increased arithmetically while the population grew geometrically. He was proven wrong because the food supply had rather increased. This increase in food supply was boosted by innovations in agriculture as well as the improved distribution of food. The current population growth rate of the world is 1.09% whereas the worldwide food supply is estimated to increase at 1.1%. Since the food supply is projected to increase at a rate greater than the current population growth rate, there is a debate that whether or not it is necessary to plan for future food shortages due to overpopulation over the next 10 years. This report has concluded that planning is indeed necessary. One reason for a call to planning is the fact that the population growth rate is greater than the estimated growth in food supply for lesser-developed countries excluding China and least-developed countries. Therefore, the population will exceed the available food in these two groups of countries by 2028. Moreover, the population growth rate trends of least and lesser developed countries are projected to increase further, thereby exacerbating the food supply shortage CITATION Pop17 \l 1033 (Population Division of the UN Department of Economic and Social Affairs, 2017). Furthermore, the current situation of food distribution disparity already calls for action. The distribution disparity exists even today in regions such as Africa, South America and Asia CITATION Foo181 \l 1033 (Food and Agriculture Organization of the United Nations, 2018). Therefore, there is a dire need to formulate policies keeping in view these facts. Planning will be of foremost importance to deal with the impending food shortages in these regions. The United Nations Food Distribution Agency must equip itself with all the essentials to ensure equitable distribution of food. Steps should be taken by the UN to decrease population growth rate in the Third World, increase food production further by fundings and introducing the latest techniques to increase yield per acre. Trade should be encouraged to promote the exchange of goods including food items. Moreover, with the looming threat of climate change, both the United Nations (US) and its Distribution Agency must equip themselves to minimize the adverse impact of climate change on food supply and to respond to any kind of emergency situations of food shortage. Without planning and implementing the said strategies, achieving the United Nations Sustainable Development goals will remain a distant dream.

Discussion

Findings of the UN Agency

As per the findings of the UN Agency, the world's population growth rate at present is 1.09% a year. The agency has also estimated that over the next 10 years the world-wide food supply will increase at a rate of 1.1% annually.

The table below shows estimated populations in 2018 and current estimated growth rates for different groupings of countries and for the world as a whole. The last column gives the agency’s estimate of the food supply in each region and the world in 2028, based on a 1.1% annual growth rate.

Group

Population in 2018 (billions)

Current Annual Growth Rate

Food Supply in 2028 (for billions of people)

More-developed countries

1.33

0.61%

1.48

Lesser-developed countries excluding China

3.85

1.23%

4.30

Least-developed countries

1.01

2.51%

1.27

China

1.44

0.05%

1.46

World

7.63

1.09%

8.51

Table SEQ Table \* ARABIC 1

Analysis of the Agency’s Findings

The given data shows that the estimated increase in worldwide food supply (1.1%) is greater than the current population growth rate of the world (1.09%). However, the given table shows that the population growth rate is greater than the estimated growth in food supply for the following categories of countries:

Lesser-developed countries excluding China (1.23%)

Least-developed countries (2.51%)

Therefore, if the annual growth rate of these two categories of countries remains the same until 2028, the population will exceed the available food in these regions. Although some experts are suggesting that there no need to plan for future shortages, these findings warrant that a strategy needs to be devised to deal with food shortages in these two groups of countries.

In the given table, the projected population in 2029 is not mentioned. In order to compare the population in 2028 with food supply in 2028, we need to calculate the population 2028 first.

The assumption for Making Calculations

It is assumed that the given annual population growth rate will remain the same over the next 10 years. On the basis of this assumption, the current annual population growth rate has been used to calculate the population in 2028. It is to be noted here that in following pages, the real trend of population growth rate in the least and lesser developed countries has also been discussed.

Calculations

In order to find the population of each category/grouping of countries in 2028, the following formula can be used:

Population in 2028 = Current population x (1+ growth-rate% )10

For example:

World population in 2028 = Current population of the world in 2018 x (1+1.09%)10

= 7.63 (1+1.09)10

= 8.5 billions

Group

Population in 2018 (billions)

Current Annual Growth Rate

Population in 2028

(billions)

Food Supply in 2028 (for billions of people)

More-developed countries

1.33

0.61%

1.41

1.48

Lesser-developed countries excluding China

3.85

1.23%

4.35

4.30

Least-developed countries

1.01

2.51%

1.29

1.27

China

1.44

0.05%

1.45

1.46

World

7.63

1.09%

8.50

8.51

The second last column of Table 2 shows the population of each grouping of countries, calculated on the basis of the above formula.

Table SEQ Table \* ARABIC 2: Comparison between Population and Food Supply in 2028

Observations

The table, thus, confirms that the population will exceed food supply in lesser-developed countries excluding China and least-developed countries. In lesser-developed countries, the population will be 4.35 billion while the food supply will be available for 4.3 billion. Likewise, in least-developed countries, the population will 1.29 billion whereas the food supply will be available for 1.27 people. Therefore there will be a shortage of food for 0.05 billion and 0.02 billion in lesser and least developed countries respectively.

However, since the overall food available in the world in 2028 will be for 8.51 whereas the population will be 8.50 as shown in Table 2. Hence there will be a food surplus for 0.01 billion (100 million people). Bu the difference is only marginal. Not planning food distribution will, therefore, be fatal. In order to plan distribution, the UN Distribution Agency should focus on the groups with food surplus (more developed countries and China). The surplus food of more developed countries and China will have to be transferred to the least and lesser developed countries in order to ensure food security and avoid malnutrition in the said food deficient regions.

Growth Rate Trends of Least and Lesser Developed Countries in Future

In the calculations, it was assumed that the current population growth rate of these groups of countries will remain same till 2028. However, practically that might not be the case. Therefore, this report has explained below the estimated trends of population growth rate in various regions.

The report titled World population Prospects published in 2017 by Population Division of the UN Department of Economic and Social Affairs provides estimations of future trends in population growth. According to this report, the population growth rate of the least developed countries will most likely increase in the near future. This increase will continue for several decades CITATION Pop17 \l 1033 (Population Division of the UN Department of Economic and Social Affairs, 2017).

The information provided in this report suggests that planning for future distribution of food is essential to overcome future food shortages and least and lesser developed countries. The new Sustainable Development Goals can only be met if policies are made while keeping this information in mind.

The Food Distribution Agency must keep track of the growth rate of the groups of countries and plan its distribution strategy accordingly.

Existing Distribution Disparity

The current situation already calls for action. According to the Food and Agriculture Organization of the United Nations, hunger is on the rise. The number of undernourished people has increased from 804 million in 2016 to about 821 million in 2018 CITATION Foo18 \l 1033 (Food and Agriculture Organization of the United Nations, 2018). This number has been increasing since 2014.

Most undernourished people live in Africa constituting nearly 21 percent of the total population. In South America, about 5.0 percent population remains undernourished. Although food shortage is on the decrease in Asia, the decreasing rate is slowing down over time. SDG target will remain unachievable without increasing efforts CITATION Foo181 \l 1033 (Food and Agriculture Organization of the United Nations, 2018).

Food Supply and Climate Change

It cannot be easily predicted what effect the unprecedented changes in climate will have on food supply. Food production depends on favorable climate conditions. Keeping in view the existing food distribution disparity, the likelihood of adverse impact of climate change on food supply, efforts need to be made to ensure equitable distribution of food from regions where food is in surplus to regions where food is in short supply.

According to Table 1 and Table 2, the world food supply in 2028 will be only marginally greater (by 100 million) than the actual population. Therefore the uncertainty in food supply caused by climate change again warrants serious efforts to plan strategies for dealing with emergency food shortages.

Recommendations

The Un Food Distribution Agency needs to devise a strategy to deal with the threats that have been highlighted above. Planning will be of foremost importance since the world food supply will be only marginally greater than the world population.

The UN Food Distribution Agency should equip itself to transfer food from countries with a surplus to countries suffering from food shortage. According to Table 2, least and lesser developed countries excluding China will be suffering extreme food shortages in the future. Therefore the Agency must devise a timeline for transfer of food to these regions.

Trade should be encouraged to promote the exchange of food items. The industrial sector in lesser and least developed countries must be enhanced in order to enable them to exchange them with food-surplus countries for food items.

The Agency must also increase its transportation capacity. In order to deal with natural disasters, it should equip itself to deal with food shortages in emergency situations.

Although the food produced today exceeds the population today, yet there is an alarming number of people suffering from malnutrition. Our calculations and assessment have shown that the situation may not become any better if proper measurements are not taken. Keeping in view the regional food shortages, growing population growth rates in the least and lesser developed the world and the worldwide food distribution disparity, the United Nations and Food Distribution Agency should take appropriate measures.

The United Nations must take steps to reduce the population growth rate in the least and lesser developed countries. For that purpose, educating the masses and creating awareness about contraceptives are necessary steps. Moreover, employment opportunities should be provided to people to minimize the desire to give birth to more children for earning a living. Minimizing the gender gap in education is of paramount in this context.

The United Nations should also endeavor to introduce the latest agriculture techniques such as drip irrigation, use of fertilizer and pesticides, genetically improved seeds, etc. to increase yield per acre. Steps must be taken to combat water shortage. Loss of water can be minimized by building proper storages, managing timings of irrigation and using pipelines.

The United Nations also needs to identify regions more prone to climate change in order to devise strategies to minimize their impact on food production.

Conclusion

Although the estimated food supply worldwide will be greater than the world population growth rate in 2028, there will be a shortage of food in the least and lesser developed countries (excluding China), because the growth rate of the population exceeds the estimated increase in food supply for these two groupings of countries. In fact, the population growth rate for these two groupings will increase further over time in the coming few decades, thereby exacerbating the malnutrition and food shortages in regions such as Asia, Africa, and Latin America. Furthermore, the disparity in food distribution exists even today in these regions. Supply of food depends on several factors, the climate is the most important. With unprecedented changes in climate, it cannot be accurately predicted how the adverse climate conditions will impact food production. Therefore food security is under a severe threat from the climate. All these facts make it important to meticulously plan food distribution for the future. The UN Food Distribution Agency must build its capacity to transfer food from countries with a surplus to countries with food shortages. The UN must promote education and awareness to tackle population growth rate. It should sponsor the latest techniques to enhance yield per acre. Trade should also be encouraged between the developed and under-developed countries to promote the import of food items into food-deficient countries. Without taking serious measures, Sustainable Development Goals cannot be achieved.

References

BIBLIOGRAPHY Food and Agriculture Organization of the United Nations. (2018). Food Security and Nutrition around the World. FAO.

Population Division of the UN Department of Economic and Social Affairs. (2017). World Population Prospects. UN Department of Economic and Social Affairs.

Subject: Maths

Pages: 7 Words: 2100

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