The volumes of two similar solids are 857.5mm ^3 and 540mm^ 3 The surface area of the smaller solid is 108m m ^ 2 What is the surface area of the larger solid ?
To solve this problem, we need to understand that the ratio of the volumes of two similar solids is equal to the cube of the ratio of their corresponding sides, and the ratio of their surface areas is equal to the square of the ratio of their corresponding sides.
Let's denote the ratio of the volumes of the two solids as \( k \):
\( \frac{V_2}{V_1} = k^3 \)
Given that the volumes of the two solids are 857.5mm^3 and 540mm^3, we can set up the following equation:
\( \frac{857.5}{540} = k^3 \)
\( k^3 = \frac{857.5}{540} \)
Solving for \( k \), we get:
\( k \approx 1.25 \)
Now, let's denote the ratio of the surface areas of the two solids as \( k_a \):
\( \frac{A_2}{A_1} = k_a^2 \)
Given that the surface area of the smaller solid is 108mm^2, we have:
\( \frac{A_2}{108} = 1.25^2 \)
\( A_2 = 108 \times 1.25^2 \)
\( A_2 \approx 168.75 \, mm^2 \)
Therefore, the surface area of the larger solid is approximately 168.75 mm^2.