Page 118 - How To Solve Word Problems In Calculus

4. A piece of wire 100 inches long is to be used to form a square
and/or a circle. Determine their (a) maximum and (b) minimum
combined area.
5. Find the maximum area of a rectangle inscribed in a semicircle of
radius 5 inches if its base lies along the diameter of the semicircle.
6. An open boxis to be constructed from a 12- × 12-inch piece of
cardboard by cutting away squares of equal size from the four
corners and folding up the sides. Determine the size of the cutout
that maximizes the volume of the box.
7. A window is to be constructed in the shape of an equilateral
triangle on top of a rectangle. If its perimeter is to be 600 cm, what
is the maximum possible area of the window?
8. Postal regulations require that the sum of the length and girth of a
rectangular package may not exceed 108 inches (the girth is the
perimeter of an end of the box). What is the maximum volume of a
package with square ends that meets this criteria?
9. A rectangle is inscribed in a right triangle whose sides are 5, 12, and
13 inches. Two adjacent sides of the rectangle lie along the legs of
the triangle. What are the dimensions of the rectangle of maximum
area? What is the maximum area?
10. Find the dimensions of the right circular cylinder of maximum
volume that can be inscribed in a right circular cone whose radius is
3 in and whose height is 10 in. What is the maximum volume?
11. What is the minimum amount of fencing needed to construct a
2
rectangular enclosure containing 1800 ft using a river as a natural
boundary on one side?
12. An open rectangular boxis to have a base twice as long as it is wide.
3
If its volume must be 972 cm , what dimensions will minimize the
amount of material used in its construction?
2
13. Find the points on the parabola y = x closest to the point (0, 1).
14. A publisher wants to print a book whose pages are each to have an
2
area of 96 in . The margins are to be 1 in on each of three sides and
2 in on the fourth side to allow room for binding. What dimensions
will allow the maximum area for the printed region?
3
15. A closed cylindrical can must have a volume of 1000 in . What
dimensions will minimize its surface area?
3
16. A closed cylindrical can must have a volume of 1000 in . Its lateral
surface is to be constructed from a rectangular piece of metal and
its top and bottom are to be stamped from square pieces of metal
and the rest of the square discarded. What dimensions will minimize
the amount of metal needed in the construction of the can?

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