There are many square prisms with volume 125 in. Let w represent the side length of the square base and h represent the height in inches. Then the formula for the volume V, in cubic inches of these prisms is V = w2h and the equation for the surface area S, in
square inches, is S = 2w2 + 4wh.

The volume is given by 125 = w²h while the surface area is given by S = 2w² + 4wh.

Let w represent the side length of the square base and h represent the height in  inches.

The volume of the square based prism is:

Volume = w²h

Since the volume is 125 in., hence:

125 = w²h

The surface area (S) is:

S = 2w² + 4wh

The volume is given by 125 = w²h while the surface area is given by S = 2w² + 4wh.

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5in by 5in by 5in

Step-by-step explanation:

We are not told wat to find but we can as well find the dimension of the prism that will minimize its surface area.

Given

Volume = 125in³

Formula

V = w²h 1

S = 2w²+4wh 2

w is the side length of the square base

h is the height of the prism

125 = w²h

h = 125/w² 3

Substitute eqn 3 into 2 as shown

S = 2w²+4wh

S = 2w²+4w(125/w²)

S = 2w²+500/w

To minimize the surface area, dS/dw = 0

dS/dw =4w-500/w²

0= 4w-500/w²

Multiply through by w²

0 = 4w³-500

-4w³ = -500

w³ = 500/4

w³ =125

w = cuberoot(125)

w = 5in

Get the height

125 =w²h

125 = 25h

h = 125/25

h = 5in

Hence the dimension of the prism is 5in by 5in by 5in

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