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Math 75: Exam 4. Take-home review for Final

You must turn in your exam at the beginning of class May 13th. If you are late toclass I may not accept your exam and you will get a zero. Your answers should bewritten in a blue-book. They can be purchased at the bookstore.

Please write neatly and in pencil. Show all of your work. Indicate your finalanswers clearly. Indicate the problem number clearly. Give units whenever applica-ble. Every problem (or part of a problem) is worth 2 points. 2 points for perfectlycorrect work, 1 point for mostly correct work, and 0 otherwise. Therefore you mustbe very careful in your work and try not to make any mistakes.

1. A company that manufactures printers sells them for $240. Each printer costs $100 to make, andfixed costs for the company are $120,000.

a) Write a linear equation for the cost of manufacturing  x printers.

b) Graph the linear model you found in part a). Make sure your graph is detailed and accurate.

c) How many calculators must be made to break-even? Round up to the nearest calculator.

2. Lets consider the widget market. There we have a price-supply equation given by p(x) = 0.25x−7where  p  is in $ and  x is the number of widgets.

a) Suppose the price of a widget is $10 when demand is 50 widgets, and the price is $6 whendemand is for 70 widgets. If we assume that price and demand are linearly related, then findthe price-demand function.

b) Find the equilibrium point for widgets. Give the number of widgets and the price.

3. Jeff deposited $5000 into an IRA that earns 0.425% per quarter. How many years until the moneygrows to $30,000? Round up to nearest year.

4. If Jeff (from problem 3) wants his $5000 deposit to become $30,000 in 5 years, what rate, com-pounded annually, would he need to find?

5. In a new housing development, the houses are selling for $100,000 and require a 20% downpayment on a 30-year mortgage, at 4% compounded monthly.

a) What are the monthly payments?

b) What is the total interest paid over 30 years?

6. I sell my car to a neighbor for $2000, and let them finance it at 7% simple interest over the next36 months. 8 months into the loan I sell the note (the loan contract) to my dentist for $2100.

a) What annual simple interest rate did I earn?b) What annual simple interest rate did my dentist earn?

7. Consider the cost function C (x) = 48, 000+1500x for a computer company, where x  is the numberof computers, and 1000 ≤ x ≤ 60, 000.

a) What is the exact cost of the 5000th computer?

b) Use marginal cost to estimate the cost of the 5000th computer?

8. A business borrows $800,000 at 0.785% per month for 10 years to buy office space. At the end of the 2nd year, how much equity do they have in the office space, if it would net $1.05 million onthe open market.

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9. The research department in a company that makes clocks has found the following price-demandand cost functions:

 p(x) = 50− 0.00125x C (x) = 160 + 0.01x   1000 ≤ x ≤ 40, 000

where  x  is the number of clocks,  p(x) is in dollars and  C (x) is in thousand dollars.

a) Find the profit function.b) For what quantities of clocks will there be a profit (as oppossed to a loss).

c) Find the marginal profit function.

d) Graph the profit function. (Be detailed and accurate.)

e) Calculate and interpret P (25) in a complete sentence.

f) Find the maximum profit. (Make sure you actually give the profit.)

10. Consider the cost function C (x) = 48, 000+1500x for a computer company, where x  is the numberof computers, and 1000 ≤ x ≤ 60, 000.

a) What is the exact cost of the 5000th computer?b) Use marginal cost to estimate the cost of the 5000th computer?

11. Mary deposits $5,000 annually for 10 years, into an account paying 6% compounded annually.After her last deposit, she stops depositing money, but lets her account sit for an additional 20years.

a) How much money is in the account at the end (20 years after she stopped depositing)?

b) Mary’s partner, Chris, decides to start making annual deposits into an account that earns thesame as Mary’s account. If Chris makes deposits for 10 years, how much should deposits be,so that Chris has the same amount of money in the end as Mary will?

12. Kellen wants to start depositing money annually into an IRA that pays 2.5% compounded annu-ally. He plans on depositing for 50 years. He then hopes to live for an additional 100 years afterretirement. If he plans on withdrawing $25,000 per year during his retirement, how much shouldhe start depositing annually today?

13. The price-demand equation for home-delivered large pizzas is   p + 0.002x  = 38.2 where   x   is thenumber of pizzas. The current price of one pizza is $21.

(a) What is the elasticity of demand function, E ( p)?

(b) Given a 10% price increase what is the approximate relative change in demand?

(c) Given the 10% price increase does the revenue increase or decrease? Indicate whether it is

elastic, inelastic, or unit demand.

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