Select the correct answer.

Cube A and cube B are similar solids. The volume of cube A is 27 cubic inches, and the volume of cube [tex]$B[tex]$[/tex] is 125 cubic inches. How many times larger is the base area of cube [tex]$[/tex]B[tex]$[/tex] than the base area of cube [tex]$[/tex]A$[/tex]?

A. [tex]\frac{25}{9}[/tex]
B. [tex]\frac{5}{3}[/tex]
C. [tex]\frac{125}{27}[/tex]



Answer :

To determine how many times larger the base area of cube B is than the base area of cube A, we start by understanding the relationship between the volumes, side lengths, and areas of similar solids.

Given the volumes of the cubes:
- Volume of cube A (\(V_A\)) is 27 cubic inches.
- Volume of cube B (\(V_B\)) is 125 cubic inches.

1. Determine the side lengths of the cubes:

To find the side length of each cube, we use the formula for the volume of a cube \(V = s^3\), where \(s\) is the side length.

- For cube A:
[tex]\[ V_A = 27 \implies s_A^3 = 27 \implies s_A = \sqrt[3]{27} = 3 \, \text{inches} \][/tex]

- For cube B:
[tex]\[ V_B = 125 \implies s_B^3 = 125 \implies s_B = \sqrt[3]{125} = 5 \, \text{inches} \][/tex]

2. Find the ratio of the side lengths:

- The side length of cube A (\(s_A\)) is 3 inches.
- The side length of cube B (\(s_B\)) is 5 inches.
- The ratio of the side lengths is:
[tex]\[ \frac{s_B}{s_A} = \frac{5}{3} \][/tex]

3. Determine the ratio of the base areas:

The base area of a cube is proportional to the square of its side length. Therefore, the ratio of the base areas of cube B to cube A is the square of the ratio of their side lengths.

- The ratio of the base areas is:
[tex]\[ \left(\frac{s_B}{s_A}\right)^2 = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \][/tex]

Thus, the base area of cube B is \(\frac{25}{9}\) times larger than the base area of cube A.

So, the correct answer is:
A. [tex]\(\frac{25}{9}\)[/tex]