To determine the surface area of the larger solid, we need to follow these steps:
1. Identify the ratio of the volumes: Since the solids are similar, the ratio of their volumes will be equal to the cube of the ratio of their corresponding side lengths. Given the volumes:
- Volume of larger solid = 729 inches³
- Volume of smaller solid = 125 inches³
2. Calculate the ratio of the side lengths:
The ratio of the volumes is given by:
[tex]\[
\text{Ratio of volumes} = \frac{\text{Volume of larger solid}}{\text{Volume of smaller solid}} = \frac{729}{125}
\][/tex]
The side length ratio will be the cube root of the volume ratio:
[tex]\[
\text{Ratio of side lengths} = \left(\frac{729}{125}\right)^{1/3} = 1.8
\][/tex]
3. Determine the ratio of the surface areas:
The surface area ratio of similar solids is the square of the ratio of their corresponding side lengths. Therefore:
[tex]\[
\text{Ratio of surface areas} = (1.8)^2 = 3.24
\][/tex]
4. Calculate the surface area of the larger solid:
We know the surface area of the smaller solid is 74.32 inches². Using the surface area ratio, the surface area of the larger solid can be calculated by multiplying the surface area of the smaller solid by the surface area ratio:
[tex]\[
\text{Surface area of larger solid} = \text{Surface area of smaller solid} \times \text{Ratio of surface areas} = 74.32 \times 3.24 = 240.80 \text{ inches}^2
\][/tex]
5. Final Answer:
Thus, the surface area of the larger solid is 240.80 square inches, so:
240.80 in. [tex]$^2$[/tex] is the correct answer.
