The volumes of two similar solids are 729 inches [tex]\(^3\)[/tex] and 125 inches [tex]\(^3\)[/tex]. If the surface area of the smaller solid is 74.32 inches [tex]\(^2\)[/tex], what is the surface area of the larger solid? Round to the nearest hundredth.

A. 133.78 in.[tex]\(^2\)[/tex]

B. 240.80 in.[tex]\(^2\)[/tex]

C. 433.43 in.[tex]\(^2\)[/tex]

D. 678.32 in.[tex]\(^2\)[/tex]



Answer :

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.