Honors 2201-4
Homework 14 & 15: due Tuesday December 2 (worth 40 points)
(from Applied Calculus, Hughes-Hallett et.al.)
Fall 2003
(1) Revenue is given by R(q) = 450q and cost is given by C(q) = 10000 + 3q 2 . For what
quantity q is profit maximized? What is the total profit at this production level?
(2) When production is 2000, marginal revenue is $4 per unit and marginal cost is $3.25 per
unit. Will maximum profit occur at a production level above or below 2000? Why? (You
should not be doing a lot of algebra for this problem.)
(3) At a price of $8 per ticket, a musical theater group can fill every seat in the theater,
which has a capacity of 1500. For every additional dollar charged, the number of people
buying tickets decreases by 75. What ticket price maximizes revenue? Explain.
(4) You run a small furniture business. You sign a deal with a customer to deliver up to 400
chairs, the exact number to be determined by the customer later. The price will be $90 per
chair up to 300 chairs, and above 300, the price will be reduced by $0.25 per chair for every
additional chair over 300 ordered. The reduced price will be used to calculate the cost of the
entire order, not just on the chairs over 300. What is the largest revenue your company can
make under this deal? Explain each step you are taking. Don’t forget to consider both upper
and lower endpoints. This problem is similar to the apple orchard problem from homework
13. Notice that the profit function you get should be continuous.
(5) Cost and revenue functions are shown in the figure labeled Problem 5.
(a) For what production levels is the profit function positive? Indicate on the graph the
endpoints of the intervals you found. Explain.
(b) Estimate the production level at which profit is maximized for the interval 0 ≤ q ≤ 300?
On the graph, draw the appropriate line. Explain.
(6) The graph of a cost function is given in the figure labeled Problem 6.
(a) At q = 100 estimate both the average cost and the marginal cost. Draw the appropriate
lines on the graph showing where your answers came from.
(b) If the manufacturer wants to decrease the average cost, should q be increased or decreased
from q = 100? Use your result from part (a) and explain your answer.
(c) Estimate the value of q that minimizes the average cost over the interval [0, 300]. Draw
the appropriate line on the graph.
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(7) The cost of producing q items is C(q) = 2500 + 12q dollars
(a) What is the marginal cost of producing the 100th item? the 1000th item?
(b) What is the average cost of producing 100 items? 1000 items? What is the average cost
as the number produced becomes very, very large (q >> 2500)?
(8) The marginal cost at a production level of 2000 units of an item is $10 per unit and the
average cost of producing 2000 units of an item is $15 per unit. If the production level were
increased slightly above 2000, would the average cost increase or decrease, or is it impossible
to tell? How do you know? (You should not be doing a lot of algebra for this problem.)
(9) You are the manager of a firm that produces slippers that sell for $20. You are producing
1200 slippers each month, at an average cost of $2 per slipper. The marginal cost at a
production level of 1200 is $3 per slipper. Assume that you sell all the slippers produced
each month.
(a) Find the general equation for revenue as a function of q. What is R(1200)?
(b) Calculate the monthly profit when q = 1200. Are you making or losing money?
(c) Will increasing production increase or decrease your average cost? Your profit? Why?
Remember the nice relationships between marginal revenue, marginal cost and average cost.
(d) Would you recommend that production be increased or decreased? Why?
(10) What is the effect of a 3% price increase on the demand if the elasticity of the good is
E = 2? E = 0.5?
(11) Determine if the elasticity of demand for the given item is elastic or inelastic. Explain.
(a) a high definition television set
(b) a particular brand of laundry detergent given that there are many brands available
(c) local telephone service given that there is only one company offering local service
(12) The demand for yams is given by q = 5000 − 10p2 , where q is in pounds of yams and
p is the price of a pound of yams.
(a) If the current price of yams is $2 per pound, how many pounds will be sold?
(b) Is the demand at $2 elastic or inelastic? Is it more accurate to say “People want yams
and will buy them no matter what the price” or “Yams are a luxury item and people will
stop buying them if the price gets too high”?
(c) At a price of $2 per pound, what is the revenue for the yam farmer?
(d) Write revenue as a function of price, and then find the price that will maximize revenue.
(e) What quantity is sold at the price you found in part (d), and what is the total revenue?
(f) Show that E = 1 at the price you found in part (d).
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Problem 5
Dollar Amount
300
R(q)
C(q)
200
100
0
0
100
200
Quantity (q)
300
Problem 6
Dollar Amount
300
C(q)
200
100
0
0
100
200
Quantity (q)
3
300