Name:________________________

Learning Motivator II - Quantitative Economics 39

April 24, 1997

Check to make sure that you have five pages and nine questions for the exam. Please show all of your work clearly in the space provided. Illegible writing will be marked incorrect. If you use the back of a page, give me some arrows to help me find it. Good Luck!

1) [10 points] Find the limit if it exists:

2) [10 points] Find *f'(x)*:

a) *f(x)* = 5e3x - ln(5x-5)

b)

3) [12 points] The UPS will pick up and deliver packages whose length plus girth (ask me to explain girth if you’re unclear) does not exceed 108". Fisher Corporation ships a granular grinding abrasive in rectangular boxes with square ends. What is the largest volume of abrasive Fisher can ship in a box? (Neglect the volume of the materials of which the box is made.) Use *x* as the side of the square base and *L* as the length.

- Write the expression for the volume, V, in terms of L and x.

- Express V(x) as a function of x alone.

- What value of x maximizes V(x)? Prove this is a maximum.

- What are the dimensions of the maximum-volume box?

4) [10 points] If the total cost of producing *t* tons of tonal is

C(t) = 0.0005t^{3} - 0.3t^{2} + 100t + 30,000,

compute the marginal cost when output is 300 tons.

5) [12 points] Find the *x* coordinate for each local optimum and inflection point that exists and indicate which is which:

f(x) = x^{3} - 2x^{2} - 4x + 3

6) [12 points] In the figure above, points B, C, and D are on a horizontal line and A is 8.4 miles above B on a perpendicular to BD. Straight-line sections of road are to be constructed from A to C and then from C to D. Construction cost along BD is $200,000 per mile, but the cost above BD is $290,000 per mile. The distance BD is 15 miles.

a) How many miles from B should the intersection C be located if cost is to be minimized?

b) Compute the minimum cost.

c) How much more than the minimum would be the cost of a single segment from A to D?

7) [12 points] An oil deposit contains 1,000,000 barrels of oil, which, after being pumped from the deposit, yields a revenue of $12 per barrel. The proportion of the deposit that will have been pumped out after t years of pumping is 0.9 - 0.9e^{-0.16t}.

Operating costs are $345,000 per year.

a) How long should pumping be continued to maximize profit?

b) Compute the maximum profit.

8) [10 points] Find the value of the function at its optimum and saddle points, if any exist. State whether each point is a local maximum, a local minimum, or a saddle point.

f(x,y) = 10xy - 5x^{2} - 6y^{2} + 20x.

9) [12 points] Computer Kate has an America On-Line (AOL) account that provides 5 hours of connection time per month. She must divide her time between hours in the chat rooms (C) and hours on the Internet (I). Her utility function is U = CI-I^{2}. How many hours should she allocate to each of the AOL services?