The base of a rectangular box is to be twice as long as it is wide. the volume of the box is 864 cubic inches. the material
for the top costs $0.50 per square inch and the material for the sides and bottom costs $0.25 per square inch. find the
dimensions that will make the cost a minimum.

find the

length=
width=
height=

length = 12 inches

width = 6 inches

height = 12 inches

Step-by-step explanation:

If the length, width, and height are L, W, and H, then:

L = 2W

864 = LWH

If the cost is C, then:

C = 0.50 LW + 0.25 (LW + 2LH + 2WH)

C = 0.50 LW + 0.25 LW + 0.50 LH + 0.50 WH

C = 0.75 LW + 0.50 LH + 0.50 WH

We need to get the cost in terms of a single variable.

Substitute the first equation into the second equation to get H in terms of W:

864 = (2W) WH

864 = 2W² H

432 = W² H

H = 432 / W²

Now substitute the equations for L and H into the cost equation:

C = 0.75 (2W) W + 0.50 (2W) (432 / W²) + 0.50 W (432 / W²)

C = 1.5 W² + 432 / W + 216 / W

C = 1.5 W² + 648 / W

Take derivative and set to 0:

dC/dW = 3W − 648 / W²

0 = 3W − 648 / W²

3W = 648 / W²

3W³ = 648

W³ = 216

W = 6

Therefore:

L = 12

H = 12

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step-by-step explanation: