Question

SOS PLEASE ANSWER ASAP!! A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 96 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.) FIND X AND Y

Answer 1

• x = 16 in
• y = 32 in

Explanation:

You want the dimensions of a maximum-volume square cuboid with a combined length and girth of 96 inches.

Volume

In this scenario, the length is represented by y, and the dimension of one side of the square is represented by x. Then we want the values of x and y that maximize the volume.

4x +y = 96 . . . . . . constraint on dimensions

y = 96 -4x . . . . . . y expressed in terms of x

The volume is ...

V = x²y

V = x²(96 -4x) = -4(x³ -24x²)

Maximum

The volume will be a maximum when its derivative with respect to x is zero.

V' = -4(3x² -48x) = 0

x(x -16) = 0 . . . . . . . divide by -12

This has solutions x = 0 and x = 16. The x=0 solution is extraneous here.

y = 96 -4x = 96 -64 = 32

The dimensions for maximum volume are x = 16 in and y = 32 in.

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Additional comment

In the volume expression, the exponents of x and y are 2 and 1, respectively. You will notice that the final allocation of available length+girth resource is girth:length = 2:1. Quite often, maximization problems will demonstrate this sort of exponent-related resource allocation.

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