8.3 Optimization Problems AP Calc I

1. A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the

metal to manufacture the can.

2. Find the point on the parabola =2 2y x that is closest to the point (1, 4).

3. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.

4. The sum of two nonnegative numbers is 20. Find the numbers if the sum of their squares is to be as

small as possible.

5. What is the smallest perimeter for a rectangle whose area is 16 square inches?

6. Show that among all rectangles with an 8 foot perimeter, the one with the largest area is a square.

7. You are planning to make an open rectangular box from an 8 inch by 15 inch piece of cardboard by

cutting squares from corners and folding up the sides. What are the dimensions of the box of largest

volume you can make this way?

8. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a

single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can

enclose?

9. A 216 square meter rectangular pea patch is to be enclosed by a fence and divided into two equal parts

by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the

smallest total length of fence? How much fence will be needed?

10. The product of two positive numbers is 16. Find the numbers if the sum of one of the numbers and the

square of the other number is the least amount.

11. An open box is to be constructed from a rectangular piece of cardboard which is 12 inches long and 8

inches wide by cutting equal squares from each corner and then turning up the sides. Find the

maximum volume of such a box.

12. A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain

180,000 square meters in order to provide enough grass for the herd. What dimensions would require

the least amount of fencing if no fencing is needed along the river?

13. An open box is to be constructed from a rectangular piece of cardboard which is 24 inches long and 15

inches wide by cutting equal squares from each corner and then turning up the sides. Find the

maximum volume of such a box.

14. Suppose you are building a rectangular stock pen using 600 feet of fencing. You will use part of this

fencing to build a fence across the middle of the rectangle. Find the length and width of the rectangle

that give the maximum total area.

15. A six-room motel is to be built with the floor plan shown.

Each room will have 350 square feet of floor space.

a. What dimensions should be used for the rooms in

order to minimize the total length of the walls?

350

ft2

b. How would your answer change if the motel had 10 rooms?

c. How would your answer change if the motel had 3 rooms?

16. Ella Mentary has 600 feet of fencing to enclose two fields. One field will be a rectangle twice as long as

it is wide, and the other will be a square. The square field must contain at least 100 square feet. The

rectangular one must contain at least 800 square feet.

a. If x is the width of the rectangular field, what is the domain of x?

b. Plot the graph of the total area contained in the two fields as a function of x.

c. What is the greatest area that can be contained in the two fields?

17. A rectangular box with a square base and no top is to be constructed using a total of 120 square cm of

cardboard. Find the dimensions of the box of maximum volume.

18. You are building a glass fish tank that will hold 72 cubic feet of water. You want its base and sides to be

rectangular and the top, of course, to be open. You want to construct the tank so that its width is 5 ft

but the length and depth are variable. Building materials for the tank cost $10 per square foot for the

base and $5 per square foot for the sides. What is the cost of the least expensive tank?

19. A track of perimeter 400 m is to be laid out on the practice field. Each semicircular end must have a

radius of at least 20 m, and each straight section must be at least 100 m. How should the track be laid

out so that it encompasses the least area?

20. A ladder is to reach over a fence 8 ft high to a wall that is 1 ft behind the fence. What is the length of

the shortest ladder that you can use?

21. A nonfolding ladder is to be taken around a corner where two hallways

intersect at right angles. One hall is 7 feet wide, and the other is 5 feet wide.

bWhat is the maximum length the ladder can be so that it will pass around such

a corner, given that you must carry the ladder parallel to the floor?