Question

8. If 108 in' of material is available to make a box with a square base and an open top, find the dimensions that will produce the maximum volume of the box.

Answer 1

box area of the Let it be the base of a square and by be the height. We know that the surface boxe can be represented as a S = x² + 4xy - Given that s = 108 in 2 x² + 4x4 = 108 - 449 = 108-xt - y - 108-a? 4x y = 108 - x 27 a f 7 We know that the Volurse of the box is v=xg③ from ② & ③ we get v=x[ 27 2 - 2 ] V = 272-* to = 27 - 30² a da 4 To tind max or men, du =0 27-31 - 0

8x²=279 x² = 36 x=16 * 2 = 6 dv d.x² = 0-3 x dx² voheme. at x=6, dr =-9 <0 X=6 ů a korset dimention of box with maximum put x=6 in ③ we get in ? - 3 = = 3 = y=3 . the box are Hence the dimensions of Base = 6 inch height = 3 inches