To determine the surface area of the smaller solid, we can use the properties of similar solids. Here’s the step-by-step solution:
1. Determine the Ratio of the Volumes:
Given the volumes of the two similar solids, we have:
[tex]\[
\text{Volume of smaller solid} = 210 \, \text{m}^3
\][/tex]
[tex]\[
\text{Volume of larger solid} = 1,680 \, \text{m}^3
\][/tex]
The ratio of the volumes can be determined using the formula for the ratio of similar solids:
[tex]\[
\text{Ratio of volumes} = \left( \frac{\text{volume of smaller solid}}{\text{volume of larger solid}} \right)^{1/3}
\][/tex]
[tex]\[
\text{Ratio of volumes} = \left( \frac{210}{1680} \right)^{1/3}
\][/tex]
Simplifying inside the parentheses:
[tex]\[
\frac{210}{1680} = \frac{1}{8}
\][/tex]
Thus:
[tex]\[
\left( \frac{1}{8} \right)^{1/3} = 0.5
\][/tex]
So, the ratio of the side lengths of the similar solids is \( 0.5 \)
2. Determine the Ratio of the Surface Areas:
Since the surface area ratio is the square of the ratio of the sides:
[tex]\[
\text{Ratio of surface areas} = (0.5)^2 = 0.25
\][/tex]
3. Calculate the Surface Area of the Smaller Solid:
We know the surface area of the larger solid:
[tex]\[
\text{Surface area of larger solid} = 856 \, \text{m}^2
\][/tex]
Therefore, we can calculate the surface area of the smaller solid using the ratio of the surface areas:
[tex]\[
\text{Surface area of smaller solid} = \text{Surface area of larger solid} \times 0.25
\][/tex]
[tex]\[
\text{Surface area of smaller solid} = 856 \times 0.25 = 214 \, \text{m}^2
\][/tex]
Hence, the surface area of the smaller solid is \( 214 \, \text{m}^2 \).
Among the given options, the correct answer is:
[tex]\[
\boxed{214 \, \text{m}^2}
\][/tex]
