Name: Directions: For each question, show the specific mathematical notation that leads to your answer. Round final answers to three decimals unless the context dictates otherwise. 1. The demand for board games can be modeled by D( p) 900(0.95) p thousand games where p is the price in dollars per game. Find the consumers surplus when the market price for the board game is $5.00 per game. Show all of your work using the algebraic method. -3. Suppose overall demand for kerosene in the United States is given by D( p) 1.5p 1.5 million gallons where p is the price per gallon in dollars. Check point: D(5)=1.51 Is demand for kerosene elastic, inelastic, or unit elastic when kerosene is sold for $3.49 per gallon? Show the mathematical justification for your answer. For what interval of prices is demand for kerosene elastic? 1
3. The life expectancy (in years) of a certain brand of clock radio is a continuous random variable ( ) x 0 with probability density function f x randomly selected clock radio lasts: a. between and 5 years if x 0 otherwise. Find the probability that a b. at most 6 years c. more than 6 years d. exactly 3 years 4. Determine if each of the following could represent probability density functions. Show supporting work. a. b. m(x) 1.6 1.4 1. 1 0.8 0.6 0.4 0. 0-1 0 1 x r(t).5 1.5 1 0.5 0-1 -0.5 0 1 3 4 5-1 -1.5 - -.5 t c. g( x) 0 otherwise 3 1 4x 4 x if 0 x 1
5. The demand for the Who Wants to Be A Millionaire computer game sold online can be modeled by D( p) -0.09 p 150 thousand games where $p per game is the price for each game. a. Label the units on the axes of the above graph. b. What is the consumers willingness and ability to spend on the computer game, when 90,000 games are demanded? Shade this area on the graph and find this amount. 6. The demand for digital cameras can be modeled by d( p) -.007 p -.39 p 10 hundred cameras when the price is $p per camera. The supply of these cameras can be described 0 when p 60 by s( p) hundred cameras when the price is $p per camera. 0.006 p -.09 p - 3.1 when p 60 Sketch both graphs on the axes provided to the right. Label each of the following (with specific values when possible): d(p) s(p) the shutdown price pmax the equilibrium price & quantity the producer surplus at equilibrium the consumer surplus at equilibrium 3
7. Evaluate the improper integral values in the table. 1 100 dx 3 x using the numerical method. Report unrounded Conclusion: 1 100 dx 3 x = 8. In a certain city, the daily use of water (in hundreds of gallons) per household is a continuous.15 x.15e if x 0 random variable with probability density function f ( x). 0 otherwise Find and interpret each of the following. Shade the corresponding area on the graph. a. 10 f ( x) dx 5 b. f ( x) dx 5 4
9. Suppose that h, the total number of hours a student spends each day on Twitter is distributed 0.5h he when 0 h 4 according to the density function T ( h). 0 elsewhere a. Find the average amount of time a student spends on Twitter in a day. b. Find and interpret P( h ). c. Find the standard deviation of the total number of hours a student spends on Twitter each day. 10. 0 years ago, Colleen started investing into a retirement account at a rate of 3000 dollars per year. If the retirement account earned 3.5% annual interest, compounded continuously over the entire 0-year period, what is the current account balance? Assume that Colleen invested the same amount each year and that the income stream was continuous. a. Explain why this question is asking for the 0-year future value, NOT the present value. b. Find the 0-year future value. 11. To save for her child s college education, Raquel would like to invest 5% of her salary each year into a savings account which earns 3.5% interest, compounded continuously. Write the function R(t) that shows the continuous stream of money flowing into the savings account, assuming Raquel s salary last year was $38,000 and i. her salary remains constant over the next 18 years. ii. her salary increases by.9% per year over the next 18 years. iii. her salary increases by $500 per year over the next 18 years. 5
1. A Clemson alumnus wants to establish a fund to provide the Math Sciences department with a continuous income stream of R(t) = $5,000 per year to use for scholarships. The fund will earn interest at a rate of 4%, compounded continuously. The amount of the endowment he needs to make now to support the scholarship for T years is equal to the T-year present value of the income stream. Find the amount of the endowment he needs to make now to support the scholarship for 10 years. Round to the nearest whole dollar. 13. For the year ending March 31, 009, sales for Toshiba Corporation were $70.1 billion. Assume Toshiba invests 8.5% of their sales amount each year, beginning April 1, 009, and that those investments can earn an APR of 4.8% compounded continuously. Assuming that Toshiba s sales are projected to decrease by 1.5% per year for the next five years, find the following: a. R( t ), the function that describes the flow of the company s investments over the next five years. b. the amount invested by Toshiba over the next five years c. the five-year future value d. the five-year present value e. the amount of interest earned over the next five years 6
14. The supply for Clemson car flags can be modeled by the following: 0 when p 5 S( p) p.19(1.995) 10 when p 5 flag. hundred flags, where $p per flag is the price of a GRAPH A GRAPH B GRAPH C a. What is the market price when 15,000 flags are sold? (be careful units are in hundreds) b. What is the supply of Clemson car flags when the price of one flag is $13? c. What is the producer revenue when the price of a flag is $13? Shade this region on graph A. d. What the minimum price for which producers will supply Clemson car flags? e. What is the producer surplus when the price of a flag is $11? Shade this region on graph B. f. What is the producer willingness and ability to receive when the price of a flag is $15? Shade this region on graph C. 7
Now consider that the demand for Clemson car flags can be modeled by D( p) -9 p 155 hundred flags where $p per flag is the price of one flag. GRAPH A GRAPH B GRAPH C g. Is there a price at which there will not be a demand for Clemson car flags? If so, what is it? If not, explain. h. For what price will flags be sold at the equilibrium point? What is the quantity at this point? i. Find the following at market equilibrium: Consumer Expenditure (shade on graph A) Producer Revenue Consumer Surplus Producer Surplus (shade on graph B) Consumer Willingness and Ability to Spend Producer Willingness and Ability to Receive Total Social Gain (shade on graph C) 8
15. The research department of a steel manufacturer believes that one of the company s rolling machines is producing sheets of steel of varying thickness. The thickness is uniformly distributed with values between 150 and 00 millimeters. Any sheets less than 160 millimeters must be scrapped because they are unacceptable to buyers. a. Calculate the average thickness of the sheets produced by this machine. b. What is the probability density function? Name the function f ( x ). c. Sketch f ( x ). Show the mean on the horizontal axis. d. What percentage of the steel sheets produced by this machine have to be scrapped? e. 5% of the steel sheets produced are thicker than millimeters. 16. At a certain grocery checkout counter, the waiting times (in minutes) follow an exponential 0.4 x 0.4e if x 0 distribution, f ( x). 0 if x 0 a. What is the average wait time at the grocery checkout counter? b. What is the probability of waiting between and 4 minutes? c. What is the probability of waiting less than 90 seconds? d. What is the probability of waiting more than 5 minutes to checkout? 9
17. The distribution of heights for all American woman aged 18 to 4 is normally distributed with a mean height of 6.5 inches and a standard deviation of.5 inches. Use the Empirical Rule to answer the following questions. a. What percentage of American women aged 18 to 4 is between 57.5 inches and 70 inches tall? b. What percentage of American women aged 18 to 4 is less than 65 inches tall? c. What percentage of American women aged 18 to 4 is between 55 inches tall and 60 inches tall? d. What percentage of American women aged 18 to 4 is less than 55 inches tall? 18. Scores on a 100-point final exam administered to all applied calculus classes at a large university are normally distributed with a mean of 7.3 and a standard deviation of 8.65. What percentage of students taking the test had: a. Scores between 60 and 80? b. Scores of at least 90? c. Scores less than 60? d. Scores that were more than one standard deviation away from the mean? e. At what score was the rate of change of the probability density function for the scores a maximum? 10