AP C
AB
W
4.6-1
Class Examples
1. Find two positive numbers whose sum is 20 and whose product is as large as possible.
2. A farmer has 2400 ft of fencing to close off a rectangular field that borders a river. What dimensions would yield a maximum area?
3. There is 500 ft of fencing to enclose 3 adjoining rectangular pens. Find the dimensions of the entire closure that will maximize the
area. Find the area.
4. What is the smallest size for a sheet of paper that is to contain 20 square inches of printed material and have 2 inch margins at the
top and bottom and 1 inch margins on the sides?
5. An open top box is to be made by cutting small squares from corners of a 12x12 sheet of tin and bending up the sides. How large
should the squares be that are cut from the corners to make the box hold as much as possible?
6. An open box with a square base is to be constructed from 42 m 2 of material. What are the dimensions if the volume is to be
maximum?
7. Design a 1 L (1000 cm 3 ) cylindrical shaped oil can. What dimensions will use the least amount of material?
Practice Problems Day #1
1. Find two positive numbers whose sum is 16 and whose product is a maximum.
2. Find two positive numbers whose product is 192 and the sum of the first plus three times the second is a minimum.
3. The perimeter of a rectangle is 500 m. Find the dimensions if the area is to be a maximum.
4. A man has 160 ft of fencing to enclose 3 sides of rectangular field. One side does not need a fence. Determine the dimensions of
the rectangle so the man will have enclosed a maximum area.
5. A rectangular garden is to be laid out along a neighbor’s lot and is to contain 432 m 2 . If the neighbor pays for half the dividing
fence, what should the dimensions of the garden be so that the cost to the owner will be a minimum?
6. A man has 3400 m of fencing for enclosing two separate lots, one of which is to be a square and the other is a rectangle twice as
long as it is wide. Find the dimensions of each lot so that the total enclosed area shall be a minimum.
7. A sheet of paper for a poster is 30 square inches. The margins on the top and bottom are 2 inches and on the sides are 1 inch.
What are the dimensions of the printed area if it is to be a maximum?
8. An open box is made from a 16 in by 30 in piece of cardboard by cutting squares from each corner and folding up the sides. What
should the size of the squares be to obtain a box of maximum volume?
9. A manufacturer wants to make an open box having a square base and a volume of 864 cubic meters. Find the dimensions and the
surface area of the box that uses the least amount of material.
10. The volume of a can is
Practice Problems Day #2
1. Find two positive numbers where the sum of the first and twice the second is 120 if the product is a maximum.
2. What positive number exceeds its cube by the maximum amount?
3. The area of a rectangle is 100 m 2 . Find the dimensions if the perimeter is to be a minimum.
4. A farmer wishes to fence in a rectangular field which is bordered on one side by a hedge. The hedge runs in a straight line and will
serve as a fence on the one side. If the farmer has 6050 yards of fencing and allows 50 yards for ties and sagging, what should the
dimensions of the field be so the area will be a maximum?
5. The Philadelphia Zoo has 500 meters of top quality wood fencing to build 8 “rooms” for the gorillas, by first constructing a fence
around a rectangular region. This region would then be partitioned into 8 smaller areas by placing 7 fences parallel to one of the
sides. Find the dimensions that will maximize the total area.
6. A page of a book contains 24 square inches of print and has 1.5 inch margins on the top and bottom and 1 inch margins on the
sides. Find the dimensions of the page so that its area is a minimum?
7. Find the volume of the largest box that can be made from a piece of cardboard, 20 inches square, by cutting equal squares from each
corner and turning up the sides?
8. A manufacturer wants to make an open box having a square base and a surface area of 192 square centimeters. Find the dimensions
and the volume of the largest box that can be made.
9. The total surface area of a can is 384
 square inches. Find the dimensions of the can if the volume is a maximum.