Calculus I
Section 4.7 – Optimization (Numbers, Rectangles, Boxes)
1. Find two positive numbers such that the second number is the reciprocal of the first and their sum is
a minimum.
2. The sum of two nonnegative numbers is 20. Find the numbers such that:
A. the sum of their squares is to be as large
as possible.
B. the product of the numbers is as large as
possible.
3. Find two positive numbers such that the sum of the first and twice the second is 120 and the product
of the numbers is a maximum.
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Calculus I
Section 4.7 – Optimization (Numbers, Rectangles, Boxes)
4. What is the smallest possible perimeter for a rectangle whose area is 16 in2 ?
5. Given the perimeter of the rectangle is 500 m, find the dimensions that will maximize the area.
6. You have 190 ft. of fencing with which to enclose your rectangular patio. One side of the patio is the
house, so no fence is needed here. Determine the dimensions of the rectangle so that the area of the
patio is a maximum.
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Calculus I
Section 4.7 – Optimization (Numbers, Rectangles, Boxes)
7. A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What
dimensions should be used so that the enclosed area will be a maximum?
y
x
x
8. If the area of a rectangle is 100 square meters, find the dimension of the rectangle if the perimeter is a
minimum.
9. An open top box is to be made by cutting small congruent squares form the corners of a 10-by-10-in.
sheet of tin and bending up the sides. How large should the squares cut from the corners be to
maximize the volume of the box?
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Calculus I
Section 4.7 – Optimization (Numbers, Rectangles, Boxes)
10. An open box is to be made from a square piece of material 20 inches on a side by cutting small
squares from each corner and turning up the sides. Find the volume of the largest box that can be
made.
11. An open box is to be made from a rectangular piece of material by cutting equal squares from each
corner and turning up the sides. Find the height of the box of maximum volume if the material has
dimensions of 2 ft. by 3 ft.
12. A manufacturer wants to design an open box having a square base and a surface area of 108 sq. in.
What dimensions will produce a box with maximum volume?
13. An open box with a square base is to be constructed from 42 square meters of material. What are the
dimensions if the volume is to be a maximum?
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