A 1,000 gallon pool (no top) with a rectangular base will be constructed such that the length of the base is twice the width. Find the dimensions (length, width, and height) of the pool that minimize the amount of material needed to construct it.

W = 4.646 ft

L = 9.292 ft

H = 3.097 ft

Step-by-step explanation:

Let's say the L is the length, W is the width, and H is the height.

The volume of the pool is:

V = LWH

The area of the pool is:

A = LW + 2LH + 2WH

L = 2W, so substituting:

V = (2W) WH

V = 2W²H

A = (2W) W + 2 (2W) H + 2WH

A = 2W² + 4WH + 2WH

A = 2W² + 6WH

Solving for H in the first equation and substituting into the second:

H = V / (2W²)

A = 2W² + 3V/W

Find dA/dW and set to 0.

dA/dW = 4W - 3V/W²

0 = 4W - 3V/W²

4W = 3V/W²

4W³ = 3V

W = ∛ (¾V)

V = 1000 gallons or 133.7 ft³.

W = ∛ (¾ (133.7 ft³))

W = 4.646 ft

So L = 9.292 ft and H = 3.097 ft.