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]
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]
