Some Optimization Problems

An artist is planning to sell signed prints of her latest work.
If 50 prints are offered for sale, she can charge $400 each.
However, if she makes more than 50 prints, she must lower the
price of all the prints by $5 for each print in excess of the 50.
How many prints should the artist make in order to maximize
her revenue?
(Source: Calculus and Its Applications, by Goldstein,
Lay and Schneider, Section 2.7 Exercise 13.)

A rectangular corral of 54 square meters is to be fenced off
and then divided by a fence into two sections. Find the
dimensions of the corral so that the amount of fencing required
is minimized.
(Source: Calculus and Its Applications, by Goldstein,
Lay and Schneider, Section 2.6 Exercise 13.)

What dimensions (radius and height, in centimeters) should be given
to a cylindrical can so that it holds exactly 1 liter
(= 1000 cm^{3}) and so that the amount of aluminum
contained in the can is minimized?
(Source: Thomas' Calculus, by Finney,
Weir and Giordano, Section 3.5 Example 2.)

As above, we seek a cylindrical can holding 1 liter. It has come
to our attention that, in the manufacturing process, the circles
for the top and bottom of the can are cut from squares, with the
excess being thrown out as scrap. If we wish to minimize the
amount of aluminum used (accounting for the scrap as well), what
dimensions should be used?
(Source: Thomas' Calculus, by Finney,
Weir and Giordano, Section 3.5 Exercise 15.)

A furniture store expects to sell 640 sofas at a steady rate next year.
The manager of the store plans to order these sofas from the manufacturer
by placing several orders of the same size spaced equally throughout the
year. The ordering cost for each delivery is $160, and carrying costs,
based on the average number of sofas in inventory, amount to $32 per
year for one sofa. Find the inventory cost in terms of the order
quantity x and the number r of orders placed during the
year. Then determine the economic order quantity.
(Source: Calculus and Its Applications, by Goldstein,
Lay and Schneider, Section 2.6 Exercise 4.)

Foggy Optics, Inc. makes laboratory microscopes. Setting up
each production run costs $2500. Insurance costs, based on the
average number of microscopes in the warehouse, amount to $20
per microscope per year. Storage costs, based on the maximum
number of microscopes in the warehouse, amount to $15
per microscope per year. Suppose that the company expects to
sell 1600 microscopes at a fairly uniform rate throughout the
year. Determine the number of production runs that will
minimize the company's overall expenses.
(Source: Calculus and Its Applications, by Goldstein,
Lay and Schneider, Section 2.6 Exercise 7.)

The average ticket price for a concert at the opera house was $50.
The average attendance was 4000. When the ticket price was raised
to $52, attendance declined to an average of 3800 persons per
performance. What should the ticket price be in order to maximize
the revenue for the opera house? (Assume a linear demand curve.)
(Source: Calculus and Its Applications, by Goldstein,
Lay and Schneider, Section 2.7 Exercise 12.)
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This page maintained by:
Thomas L. Scofield
Department of Mathematics and Statistics,
Calvin College
Last Modified:
Wednesday, 01Mar2006 18:20:42 EST