Three adjacent faces of a right rectangular prism have surface areas of 24, 36, and 54 squ… — GRE General Test

Three adjacent faces of a right rectangular prism have surface areas of $24$ , $36$ , and $54$ square centimeters. What is the volume of the prism, in cubic centimeters?

Answer: 216 cubic centimeters

Answer

The volume of the rectangular prism is

216

cubic centimeters.

Let the edge lengths of the right rectangular prism be

l

w

, and

h

. The areas of three adjacent faces meeting at a common vertex are given by

lw = 24

wh = 36

, and

lh = 54

. Multiplying these three equations together yields

(lw)(wh)(lh) = (lwh)^2 = 24 \times 36 \times 54 = 46,656

. Since the volume

V

of a right rectangular prism is equal to

lwh

, taking the positive square root gives

V = \sqrt{46,656} = 216

cubic centimeters.

Step-by-Step Solution

Define variables for the three dimensions of the rectangular prism.

Let

l

w

, and

h

represent the length, width, and height of the prism.

The area of each rectangular face is the product of two adjacent dimensions.

Express the given face areas in terms of the dimensions.

lw = 24

wh = 36

, and

lh = 54

Adjacent faces share pairwise dimensions.

Multiply all three face area equations together.

(lw)(wh)(lh) = l^2 w^2 h^2 = (lwh)^2 = 24 \times 36 \times 54 = 46,656

Multiplying the three face areas yields the square of the prism's volume (

V = lwh

Calculate the square root of the product to determine the volume.

V = lwh = \sqrt{46,656} = 216

Taking the square root gives the volume directly without needing to solve for individual edge lengths.

Key Concept

For a right rectangular prism with adjacent face areas

A_1

A_2

, and

A_3

, the volume is given by

V = \sqrt{A_1 \times A_2 \times A_3}

Estimated Time:1m 15s

Question

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Key Concept

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