A parcel delivery service will deliver a package only if the length plus the gir

Question

A parcel delivery service will deliver a package only if the length plus the girth (distance around) does not exceed 112 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements. Maximum Volume =

Explanation / Answer

lol, girth... Let l be the length, w be the width, and h be the height. Notice that girth = 2w + 2h. So we have l + 2w + 2h < 112 If the ends are square, then w = h. So this modifies the previous equation into l + 4w < 112 We need to maximize the volume: V = lwh = lw^2 l + 4w = 112 => l = 112 - 4w So: V = (112 - 4w)w^2 = 112w^2 - 4w^3 V' = 224w - 12w^2 0 = w(224 - 12w) w = 0 or w = 224/12 = 56/3 So the volume is maximized at w = 56/3 implying that: l = 112 - 4(56/3) = 112/3 Thus the maximum volume is: V = (112/3)(56/3)(56/3) = 351232/27 = 13008.59

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