To solve for the volume of the smaller solid given the surface areas of two similar solids and the volume of the larger solid, we need to understand the relationship between the surface areas and volumes of similar solids.
1. Calculate the ratio of the sides:
Given the surface areas of two similar solids, the ratio of the sides (or linear dimensions) can be found by taking the square root of the ratio of their surface areas. The ratio of the surface areas is:
[tex]\[
\frac{169}{81}
\][/tex]
The side ratio is the square root of this ratio:
[tex]\[
\frac{13}{9}
\][/tex]
2. Determine the volume scale factor:
The volume scale factor between similar solids is the cube of the side ratio. Therefore, the volume scale factor is:
[tex]\[
\left(\frac{13}{9}\right)^3
\][/tex]
3. Set up the proportion:
To find the volume of the smaller solid, [tex]\(x\)[/tex], we set up a proportion where the volumes are related by the volume scale factor:
[tex]\[
\frac{V_{\text{large}}}{V_{\text{small}}} = \left(\frac{\text{side ratio}}{1}\right)^3
\][/tex]
Given that we know the larger volume is 124.92, we can write this as:
[tex]\[
\frac{124.92}{x} = \left(\frac{13}{9}\right)^3
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{124.92}{\left(\frac{13}{9}\right)^3}
\][/tex]
After solving this, we find:
[tex]\[
x \approx 41.45
\][/tex]
So, the correct proportion to show how to solve for the volume of the smaller solid is:
[tex]\[
\frac{13^3}{9^3} = \frac{124.92}{x}
\][/tex]
