The volumes of two similar solids are 512 cm3 and 2,197 cm3. If the smaller solid has a surface area of 960 cm2, find the surface area of the larger solid.

To solve this problem, we will use the concept of similarity between solids. The ratio of the volumes of two similar solids is related to the cube of the ratio of their corresponding linear dimensions. Similarly, the ratio of the surface areas of two similar solids is related to the square of the ratio of their corresponding linear dimensions.

### Step 1: Find the ratio of the volumes
The volumes of the two similar solids are given as 512 cm³ and 2,197 cm³. The ratio of their volumes is:
\[
\frac{V_{\text{larger}}}{V_{\text{smaller}}} = \frac{2197}{512}
\]
Let's simplify this ratio:
\[
\frac{2197}{512} = \left(\frac{h_{\text{larger}}}{h_{\text{smaller}}}\right)^3
\]
where \( h_{\text{larger}} \) and \( h_{\text{smaller}} \) represent the corresponding linear dimensions of the solids.

### Step 2: Solve for the ratio of linear dimensions
To find the ratio of the linear dimensions, we take the cube root of the volume ratio:
\[
\frac{h_{\text{larger}}}{h_{\text{smaller}}} = \sqrt[3]{\frac{2197}{512}} = \frac{\sqrt[3]{2197}}{\sqrt[3]{512}} = \frac{13}{8}
\]
Thus, the ratio of the linear dimensions is \( \frac{13}{8} \).

### Step 3: Use the ratio of linear dimensions to find the surface area ratio
The ratio of the surface areas of two similar solids is the square of the ratio of their corresponding linear dimensions:
\[
\frac{A_{\text{larger}}}{A_{\text{smaller}}} = \left(\frac{13}{8}\right)^2 = \frac{169}{64}
\]

### Step 4: Find the surface area of the larger solid
We know that the surface area of the smaller solid is 960 cm². Using the surface area ratio:
\[
\frac{A_{\text{larger}}}{960} = \frac{169}{64}
\]
Solve for \( A_{\text{larger}} \):
\[
A_{\text{larger}} = 960 \times \frac{169}{64} = \frac{960 \times 169}{64} = \frac{162,240}{64} = 2,535 \, \text{cm}^2
\]

Thus, the surface area of the larger solid is **2,535 cm²**.

Would you like more details on any step?

### Related Questions:
1. How do you calculate the surface area of similar solids using their volume ratio?
2. How is the ratio of linear dimensions related to the ratio of surface areas in similar solids?
3. Can the method of ratios be used to calculate areas or volumes for irregular shapes?
4. What are the properties of similar figures in three-dimensional geometry?
5. How do volume and surface area scale for irregular objects?

### Tip:
When comparing geometric figures, remember that volume scales by the cube of the linear dimension, and surface area scales by the square of the linear dimension.

The volumes of two similar solids are 512 cm³ and 2,197 cm³. If the smaller solid has a surface area of 960 cm², find the surface area of the larger solid.

Learn how to find the surface area of a larger solid, given the volumes and surface area of two similar solids. This problem uses the concepts of volume ratio and surface area ratio, and is suitable for students in Grades 9-11.

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