Optimization Problems (Calculus Fun)
Many application problems in calculus involve functions for which you want to find maximum or minimum values. The restrictions
stated or implied for such functions will determine the domain from which you must work. The function, together with its domain,
will suggest which technique is appropriate to use in determining a maximum or minimum value - the Extreme Value Theorem, the
First Derivative Test, or the Second Derivative Test.
1. Find 2 numbers whose sum is 20 and whose product is as large as possible.
2. A sheet of cardboard has length 25 inches and a width of 20 inches. Fold it in a way in which it will create a box with maximum
volume.
3. Chuck has 20 feet of fencing and wishes to make a rectangular fence for his dog Rover. If he uses his house for one side of the
fence what is maximum area?
4. You need to fence in a rectangular play zone for children, to fit into a right - triangular plot with sides measuring 4 m and 12 m.
What is the maximum area for this play zone?
5. A manufacturer can produce a pair of earrings at a cost of $3. The earrings have been selling for $5 per pair, and at this price,
consumers have been buying 4,000 per month. The manufacturer is planning to raise the price of the earrings and estimates that
for each $1 increase in the price, 400 fewer pairs of earrings will be sold each month. At what price should the manufacturer sell
the earrings to maximize profit?
6. Two sides of a triangle are 4 inches long. What should the angle between these sides be to make the area of the triangle as
large as possible?
7. A dune buggy is on the desert at a point A located 40 km from a point B, which lies on a long, straight road. The driver can
travel 45 km/hr on the desert and 90 km/hr on the road. The driver will win a prize if he arrives on time at the finish line (point D) in
less than 1 hour. If the distance from B to D is 28 km, is it possible for him to choose a route so that he can collect the prize?
8. A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each
person pays $60. In larger groups, everybody’s fare is reduced by $1 for each person in excess of 35. Determine the size of the
group for which the bus company’s revenue will be the greatest.
9. A manufacturer estimates that when x units of a particular commodity are produced each month, the total cost (in dollars) will be
1
C( x )  x 2  4 x  200
8
and all units can be sold at a price of p(x) = 49 - x dollars per unit. Determine the price that corresponds to the maximum profit.
10. For the cost equation in #9, when will the minimum average cost occur?
11. A rectangular box with a square base and no top is to have a volume of 108 cubic inches. Find the dimensions for the box that
require the least amount of material.
12. A right circular cylinder is inscribed in a right circular cone so that the center lines of the cylinder and the cone coincide. The
cone has a height 8 cm and radius 6 cm. Find the maximum volume possible for the inscribed cylinder.