Schaum's Outline of Theory and Problems of Beginning Calculus

180 MORE MAXIMUM AND MINIMUM PROBLEMS [CHAP. 24(b) An open box (that is, a box without a top) is to be constructed with a square base [see Fig. 24-2(b)] and isrequired to have a volume of 48 cubic inches. The bottom of the box costs 3 cents per square inch, whereas thesides cost 2 cents per square inch. Find the dimensions that will minimize the cost of the box.Let x be the side of the square bottom, and let h be the height. Then the cost of the bottom is 3x2 and thecost of each of the four sides is 2xh, giving a total cost ofThe volume is V = 48 = x2h. Hence, h = 48/x2 andwhich is to be minimized on (0, + 00). NowC = 3x2 + 4(2xh) = 3x2 + 8xhc = 3x2 + fix@ = 3x2 + -384X = 3x2 + 3fj4x-land so the critical numbers are the solutions ofNow3846~ ---x2 -O3846~ = -X2x3 = 64x=4-- dZC768- 6 - (-2)384~-~ = 6 +-> 0dx2x3for all positive x; in particular, for x = 4. By the second-derivative test, C has a relative minimum at x = 4.But since 4 is the only positive critical number and C is continuous on (0, + oo), Theorem 24.1 tells us that Chas an absolute minimum at x = 4. When x = 4,48 48h=-=-=3x2 16So, the side of the base should be 4 inches and the height 3 inches.tYFig. 24-2