# Business statistics and calculus

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Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much.

I would like for you to show me all of your work/calculations and the correct answer to each problem.

For Exercise 2, find the mode of the probability distribution with the given probability density function.

4 3 2

-0.0003422x + 0.006845x - 0.04558x + 0.1136x + 0.2448, for 0 is less than or

2. f(x) - [ 0, equal to x, which is

less than or equal to 10

otherwise

1. For Exercise 3, calculate the probability that the random variable X falls between the values of 2 and 4 (lies in the interval [2,4]).

0.2, for 0 is less than or equal to x, which is less than or equal to 5.

3. f(x) = [0, otherwise

5. Suppose you run a construction contracting business, and you are bidding for a big project. You have one major competitor. Based on past experience, you believe that your competitor's bid for the project could be anywhere between $8 million and $10 million, and that all values in between are equally likely.

(a) Define the random variable for this problem.

(b) Define the probability density function for this problem, and sketch its graph.

(c) What is the probability that your competitor's bid will be between $8.9 million and $9.3 million?

(d) To what does your calculation in (c ) correspond graphically?

6. To help decide what you want to plant in your garden and when, you want to estimate when the last frost will occur. From historical records, you determine that in your area it could be anywhere between March 28 and April 19, but that any date within the interval is about equally likely to be the last frost.

(a) Define the random variable for this problem.

(b) Define the probability density function for this problem, and sketch its graph.

(c) What is the probability that the last frost will be between March 29 and April 5?

(d) What assumptions are you making in your calculations above? Comment on them.

(e) To what does your calculation in (c) correspond graphically?

For Exercise 4, an interval ([a, b]) is specified, along with a value of n (the number of equal subintervals into which you are to subdivide the interval). For the exercise, first find the value of triangle x, then find all the endpoints of the subintervals, identifying them symbolically (using the variable subscripts). If you can, also write out a general expression for the ith endpoint, x If a variable other than x is specified, adjust your

i.

answers in appropriately.

4. a = 4, b = 6; n = 8; independent variable is q.

23. You have been biking in a very flat area of Ohio, with virtually no traffic. You have a speedometer on your bike. As an exercise, for the last 2 hours, you worked to maintain a steady speed of 16 miles per hour (mph).

(a) How far did you travel? Show how units can help to understand this calculation.

(b) Draw in your velocity (speed) curve on a graph, and show what your calculation in (a) corresponds to on that graph.

24. You are stuffing envelopes for a mailing by a campus group you belong to. You estimate that you can stuff about 6 envelopes in a minute. Suppose you work for half an hour at this rate.

(a) How many envelopes would you expect to stuff? Show how units can help to understand this calculation.

(b) On a graph, sketch the graph of your rate (in envelopes per minute) versus time (in minutes). To what does your calculation in (a) correspond?

For Exercises 1 and 2, use the idea of finding the limit of a sum using a calculator program or spreadsheet template, for n = 4, 8, ..., and showing the calculations to find the definite integral of the given function f(x) over the given interval [a, b], using the given method of approximation:

1. f(x) = 4, for 1 is less than or equal to x, which is less than or equal to 5; Trapezoids.

2. f(x) = 3x + 2, for 0 is less than or equal to x, which is less than or equal to 8; Right Rectangles (to 2 significant figures)

7. Suppose you are biking in Ohio, and your velocity (in mph) after t hours of biking is given by:

v(t) = 14 + 2t, for 0 is less than or equal to t, which is less than or equal to 2.

(a) Which method(s) of approximation should give an exact answer?

(b) Show on a graph what the calculations using the Right Rectangle method with n = 2 and n = 4 correspond to, and use the graph to explain which is better.

(c) Use the calculator program or spreadsheet template to find the Right Rectangle approximation for n = 4, 8 , 16, 32, ... Put the answers into a table, and make a separate column to round off your answers to 3 significant figures. Go as far as you need to, in order to convince yourself of the correct value of the limit, rounded off to 3 significant figures. This is what we mean by finding the limit of the sums. Write a mathematical expression using the "lim" notation to express your answer.

(d) What is the interpretation of your answer in (c)?

(e) Try doing what you did in (c), but this time use Trapezoids.

8. Suppose you are stuffing envelopes and your rate (in envelopes per minute) is given by the function:

2

t

r(t) = 6 +---- , for 0 is less than or equal to t, which is less than or equal to 90.

4050

(a) Which method of approximation should give an exact answer?

(b) Show on a graph what the calculations using the Right Rectangle method with n = 2 and n = 4 corresponds to, and use the graph to explain which is better.

(c) Use the calculator program or spreadsheet template to find the Right Rectangle approximation for n = 4, 8 , 16, 32, ... Put the answers into a table as was done in this section and make a separate column to round off your answers to 3 significant figures. Go as far as you need to in order to convince yourself of the correct value of the limit, rounded off to 3 significant figures. This is what we mean by finding the limit of sums. Write a mathematical expression using the "lim" notation to express your answer.

(d) What is the interpretation of your answer in (c)?

(e) Try doing what you did in (c), but this time use Trapezoids.

(f) Find the limit of sums using Simpson's Rule as well. Which method seems to get you the answer the fastest?

For Exercises 1 and 2 below, find the indefinite integrals.

3

1. [(x + 2x)dx

2

2. [(-2x + 5)dx

14. Suppose you are stuffing envelopes and your rate (in envelopes per minute) is given by the function:

2

t

r(t) = 6 + ---- , for 0 is less than or equal to t, which is less than or equal to 90.

4050

(a) Suppose E(t) is defined to be the total number of envelopes you have stuffed after t minutes of work. What would be another notation for r(t), at least approximately? Mathematically, what are we really looking for if we want to find E(t) from r(t)?

(b) Find the most general possible form for E(t). What mathematical notation expresses this?

(c) For what specific value of t do you know the value of E(t)? Use this known fact to find the value of the constant (C) in your general expression for E(t).

(d) Now find the value of E(60). What does this mean in the context of the problem?

(e) How many envelopes did you stuff in the first half-hour of working? How could this be expressed symbolically?

(f) How many envelopes did you stuff in the second half-hour of working (between time t = 30 and time t = 60)? How could your answer be expressed symbolically?

For Exercises 1 and 2 below, evaluate the following definite integrals using the Fundamental Theorem of Calculus:

1.

[3 3

(x + 2x)dx

0

2.

[4 2

(-2x + 5)dx

2

12. As in previous exercises, suppose you are stuffing envelopes and your rate (in envelopes per minute) is given by the function:

2

t

r(t) = 6 +------, for 0 is less than or equal to t, which is less than or equal to 90.

4050

(a) How many envelopes did you stuff in the first half-hour of working? How could this be expressed symbolically using the integral notation?

(b) How many envelopes did you stuff in the second half-hour of working (between time = 30 and time = 60)? How could your answer be expressed symbolically using integral notation?

(c) Use the Fundamental Theorem of Calculus directly to find the total number of envelopes stuffed in the first full hour of working. How does this relate to your answers in (a) and (b)?

9. You are in the middle of a lawsuit. You have had some experience with similar suits, and estimate that the eventual court settlement will result in a net benefit to you of X hundred thousand dollars, with a probability density function given by:

2

f(x) = [-0.0208x + 0.0417x + 0.0208, for -2 is less than or equal to x,

0, which is less than or equal to 4

otherwise

(a) Write an integral equation with a variable upper limit whose solution would give you the median result of the court settlement.

(b) Evaluate the integral in (a) to get an equation not involving an integral whose solution would give you the median.

(c) Solve the equation in (b) to find the median.

(d) Now use technology to solve the integral equation in (a) without evaluating the integral first. Do you get the same answer? What step in the 12-Step program for Plugaholics does this represent?

(e) Can you find the 25th and 75th percentile values (the quartiles)?

10. You are waiting for a bus. From past experience, you have figured that the amount of time you have to wait in minutes, X, has the following distribution:

-0.1x

f(x) = [0.1e , for x is greater than or equal to 0

0, otherwise

(a) Write an integral equation with a variable upper limit whose solution would give you the median waiting time for the bus.

(b) Evaluate the integral in (a) to get an equation not involving an integral whose solution would give you the median.

(c) Solve the equation in (b) to find the median. Can you verify your solution in a different way (for example, one with technology and the other without)?

(d) Now use technology to solve the integral equation in (a) without evaluating the integral first. Do you get the same answer?

(e) Can you find the 25th and 75th percentile values (the quartiles)?

For Exercises 1 and 2, find the equilibrium price and quantity, and the Consumer and Producer Surplus for the given supply and demand functions.

1. D(q) = 224 - 8q, S(q) = 36 + 12q

2. D(q) = 75 - 20q, S(q) = 40 + 8q

8. Suppose that demand and supply values for unicycles in a year can be estimated as follows:

Number of Unicycles (u) 100 200 300 400 500

Price ($) Where Demand is u 200 160 130 94 72

Price ($) Where Supply is u 50 60 72 85 100

(a) Fin models for the supply and demand functions, and justify your choices.

(b) Find the equilibrium price and quantity using your models.

(c) Find the Consumer Surplus and Producer Surplus at the equilibrium price and quantity using your models, and explain in words what each means.

(d) What would be the actual collective revenue of the producers at the equilibrium price and quantity?

(e) If unicycle makers colluded to fix the price at $130, what would happen? What would their actual collective revenue be? Would they want to do it? How can a society deal with this situation?

(f) What would be a way to get a different estimate of your answer to (e)? What do you get? What margin of error does this suggest?

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##### Solution Summary

There area variety of business-related statistics and calculus problems here, covering topics such as random variables, probability, probability density functions, velocity, integrals, methods of approximation, the fundamental theorem of calculus, supply and demand, and margin of error.

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For Exercise 2, find the mode of the probability distribution with the given probability density functions.

4 3 2

-0.0003422x + 0.006845x - 0.04558x + 0.1136x + 0.2448, for 0 is less than or

2. f(x) = [0, equal to x, which is

less than or equal to 10

Otherwise

Solution: Mode for a continuous probability distribution is obtained, when the p(x) attains maximum value in the interval [0, 10]

Find the first and second derivatives of p(x).

Then equation p'(x) = 0, you get the x =a [say] and if this value lies in [0, 10], then this is the mode for the function and also check if this value gives maximum value for the p(x).

For Exercise 3, calculate the probability that the random variable X falls between the values of 2 and 4 (lies in the interval [2,4]).

0.2, or 0 is less than or equal to x, which is less than or equal to 5.

3. f(x) = [0, otherwise

Solution: given f(x) = 0.2, 0 <= x< = 5,

0, otherwise.

We need to compute 5

P (0 < X < 5) = 0.2dx = 0.2[5 - 0] = 1.

0

Clearly shows that f(x) is a probability density function for the given r.v X.

5. Suppose you run a construction contracting business, and you are bidding for a big project. You have one major competitor. Based on past experience, you believe that your competitor's bid for the project could be anywhere between $8 million and $10 million, and that all values in between are equally likely.

(a) Define the random variable for this problem.

Solution: The random variable X follows uniform distribution (rectangular distribution). ...

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