Optimization Problems

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

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.


Answer List:

1.  10, 10; product = 100

2.  Cutting out a square with length 3.681 inches from each corner, Max Volume = 820.528 cubic inches

3.  dimension = 5 by 10 feet, max area = 50 square feet

4.  dimension = 2 by 6 m, max area = 12 m2

5.  price = 9$, max profit = 14400$

6.  90 degree, max area = 8 in2

7.  min time = 1.081 hours, cannot win prize

8.  47 or 48 people, revenue = 2256$

9.  produce 20 units, max profit = 250

10.  when producing 40 units, min average coast is 14

11.  dimension = 6 by 6 by 3 inches, min surface area = 108 in2

12.  dimension of the cylinder is with radius 4 and height 8/3 cm, max volume = 134.031 cm3