To determine the surface area of the larger solid, we need to follow a series of steps that take into account the properties of similar solids. Similar solids have proportional dimensions, and their volumes and surface areas follow certain ratios.
1. Calculate the ratio of the volumes:
The volumes of the two solids are given as 729 cubic inches and 125 cubic inches. The ratio of the volumes is:
[tex]\[
\text{Volume Ratio} = \frac{\text{Volume of Larger Solid}}{\text{Volume of Smaller Solid}} = \frac{729}{125} = 5.832
\][/tex]
2. Determine the ratio of the surface areas:
Since the solids are similar, the ratio of their surface areas is the volume ratio raised to the power of [tex]\( \frac{2}{3} \)[/tex]. This is because the surface area ratio of similar solids is proportional to the square of the ratio of their corresponding linear dimensions, and the volume ratio is the cube of the ratio of their corresponding linear dimensions.
[tex]\[
\text{Area Ratio} = (\text{Volume Ratio})^{\frac{2}{3}} = (5.832)^{\frac{2}{3}} \approx 3.24
\][/tex]
3. Calculate the surface area of the larger solid:
Given that the surface area of the smaller solid is 74.32 square inches, we can find the surface area of the larger solid by multiplying the surface area of the smaller solid by the ratio of the surface areas.
[tex]\[
\text{Surface Area of Larger Solid} = \text{Surface Area of Smaller Solid} \times \text{Area Ratio} = 74.32 \times 3.24 \approx 240.80 \, \text{square inches}
\][/tex]
Thus, the surface area of the larger solid, rounded to the nearest hundredth, is 240.80 square inches. Hence, the correct answer is:
[tex]\[
\boxed{240.80 \, \text{in}^2}
\][/tex]
