To determine how many times larger the base area of cube [tex]\( B \)[/tex] is compared to the base area of cube [tex]\( A \)[/tex], we follow these steps:
1. Understand the relationship between the volumes and linear dimensions (side lengths) of similar solids:
For similar solids, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions.
2. Calculate the ratio of the linear dimensions:
Given:
- Volume of cube [tex]\( A \)[/tex] = 27 cubic inches
- Volume of cube [tex]\( B \)[/tex] = 125 cubic inches
Let the side length of cube [tex]\( A \)[/tex] be [tex]\( a \)[/tex] and the side length of cube [tex]\( B \)[/tex] be [tex]\( b \)[/tex].
The volume of a cube is given by the side length cubed.
Therefore,
[tex]\[
a^3 = 27 \implies a = 3
\][/tex]
[tex]\[
b^3 = 125 \implies b = 5
\][/tex]
The ratio of their linear dimensions (side lengths) is:
[tex]\[
\frac{b}{a} = \frac{5}{3}
\][/tex]
3. Determine the ratio of the base areas:
The base area of a cube is the square of its side length.
For similar solids, the ratio of their base areas is the square of the ratio of their corresponding linear dimensions.
Thus, the ratio of the base areas is:
[tex]\[
\left( \frac{b}{a} \right)^2 = \left( \frac{5}{3} \right)^2 = \frac{25}{9}
\][/tex]
So, the base area of cube [tex]\( B \)[/tex] is [tex]\(\frac{25}{9}\)[/tex] times larger than the base area of cube [tex]\( A \)[/tex].
The correct answer is A. [tex]\(\frac{25}{9}\)[/tex].
