Given that the solids are mathematically similar, the ratio of their volumes is related to the cube of the ratio of their corresponding linear dimensions.

The formula for the ratio of volumes of similar solids is:

$\frac{V_1}{V_2} = \left( \frac{L_1}{L_2} \right)^3$

where $V_1$ and $V_2$ are the volumes of the smaller and larger solids, and $L_1$ and $L_2$ are their corresponding linear dimensions.

From the given ratio of volumes $\frac{V_1}{V_2} = \frac{27}{64}$, we can find the ratio of the linear dimensions:

$\frac{L_1}{L_2} = \left( \frac{27}{64} \right)^{\frac{1}{3}} = \frac{3}{4}$

The ratio of surface areas of two similar solids is the square of the ratio of their corresponding linear dimensions:

$\frac{A_1}{A_2} = \left( \frac{L_1}{L_2} \right)^2$

Using the ratio $\frac{L_1}{L_2} = \frac{3}{4}$, we find the ratio of surface areas:

$\frac{A_1}{A_2} = \left( \frac{3}{4} \right)^2 = \frac{9}{16}$

This means the surface area of the smaller solid is to the surface area of the larger solid as 9 is to 16. Now, if the surface area of the smaller solid is 117 cm², we can find the surface area of the larger solid:

$\frac{A_1}{A_2} = \frac{9}{16}$

$\frac{117}{A_2} = \frac{9}{16}$

Cross-multiply to solve for $A_2$:

$117 \times 16 = 9 \times A_2$

$1872 = 9 \times A_2$

$A_2 = \frac{1872}{9} = 208 \text{ cm}^2$

Thus, the surface area of the larger solid is 208 cm².

Would you like more details on the process or have other questions?

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1. How does the relationship between volume and surface area in similar solids work?
2. Can you explain why the ratio of surface areas is the square of the ratio of linear dimensions?
3. How would this process change if the ratio of volumes was different?
4. What happens if the solids are not similar, how do you find their surface areas and volumes?
5. Can we use this method for solids other than those with regular shapes?

Tip: When dealing with similar solids, always remember that volume ratio is linked to the cube of the linear dimensions, while surface area ratio is linked to the square of the linear dimensions.