Answer :
To find the surface area of the larger solid, we need to follow several steps carefully. Let’s go through each step of the process in detail:
1. Determine the Ratio of Volumes:
The volumes of the two similar solids are given as 729 cubic inches and 125 cubic inches. To find the ratio of their sizes, we first calculate the ratio of their volumes and then take the cube root of this ratio.
[tex]\[ \text{Ratio of volumes} = \left( \frac{\text{Volume of larger solid}}{\text{Volume of smaller solid}} \right)^{1/3} \][/tex]
Therefore,
[tex]\[ \text{Ratio of volumes} = \left( \frac{729}{125} \right)^{1/3} \approx 1.80 \][/tex]
2. Calculate the Ratio of Surface Areas:
The surfaces areas of similar solids are proportional to the square of the ratio of their corresponding linear dimensions. Therefore, we square the ratio we found.
[tex]\[ \text{Ratio of surface areas} = (1.80)^2 = 3.24 \][/tex]
3. Calculate the Surface Area of the Larger Solid:
We are given that the surface area of the smaller solid is 74.32 square inches. To find the surface area of the larger solid, we multiply the surface area of the smaller solid by the ratio of the surface areas.
[tex]\[ \text{Surface area of larger solid} = \text{Surface area of smaller solid} \times \text{Ratio of surface areas} \][/tex]
[tex]\[ \text{Surface area of larger solid} = 74.32 \, \text{in}^2 \times 3.24 \approx 240.80 \, \text{in}^2 \][/tex]
4. Round to the Nearest Hundredth:
The computed surface area of the larger solid is approximately 240.80 square inches.
Thus, the surface area of the larger solid, rounded to the nearest hundredth, is [tex]\( 240.80 \, \text{in}^2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{240.80 \, \text{in}^2} \][/tex]
1. Determine the Ratio of Volumes:
The volumes of the two similar solids are given as 729 cubic inches and 125 cubic inches. To find the ratio of their sizes, we first calculate the ratio of their volumes and then take the cube root of this ratio.
[tex]\[ \text{Ratio of volumes} = \left( \frac{\text{Volume of larger solid}}{\text{Volume of smaller solid}} \right)^{1/3} \][/tex]
Therefore,
[tex]\[ \text{Ratio of volumes} = \left( \frac{729}{125} \right)^{1/3} \approx 1.80 \][/tex]
2. Calculate the Ratio of Surface Areas:
The surfaces areas of similar solids are proportional to the square of the ratio of their corresponding linear dimensions. Therefore, we square the ratio we found.
[tex]\[ \text{Ratio of surface areas} = (1.80)^2 = 3.24 \][/tex]
3. Calculate the Surface Area of the Larger Solid:
We are given that the surface area of the smaller solid is 74.32 square inches. To find the surface area of the larger solid, we multiply the surface area of the smaller solid by the ratio of the surface areas.
[tex]\[ \text{Surface area of larger solid} = \text{Surface area of smaller solid} \times \text{Ratio of surface areas} \][/tex]
[tex]\[ \text{Surface area of larger solid} = 74.32 \, \text{in}^2 \times 3.24 \approx 240.80 \, \text{in}^2 \][/tex]
4. Round to the Nearest Hundredth:
The computed surface area of the larger solid is approximately 240.80 square inches.
Thus, the surface area of the larger solid, rounded to the nearest hundredth, is [tex]\( 240.80 \, \text{in}^2 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{240.80 \, \text{in}^2} \][/tex]
