To determine the surface area of the smaller solid given the volumes and the surface area of the larger solid, we need to follow these steps:
1. Understand the relationship of similar solids:
- For similar solids, the ratio of their volumes is equal to the cube of the scale factor (k) between corresponding linear dimensions. That is:
[tex]\[
\left( \frac{\text{Volume of small solid}}{\text{Volume of large solid}} \right) = k^3
\][/tex]
2. Calculate the scale factor:
- Given that the volume of the smaller solid is [tex]\(210 \, \text{m}^3\)[/tex] and the volume of the larger solid is [tex]\(1680 \, \text{m}^3\)[/tex]:
[tex]\[
\left( \frac{210}{1680} \right) = k^3
\][/tex]
- Simplifying the fraction, we get:
[tex]\[
\frac{210}{1680} = \frac{1}{8} = \frac{1}{2^3} = (0.5)^3
\][/tex]
- Therefore, the scale factor [tex]\(k\)[/tex] is:
[tex]\[
k = 0.5
\][/tex]
3. Relate the surface areas of the similar solids:
- The ratio of their surface areas is equal to the square of the scale factor. That is:
[tex]\[
\left( \frac{\text{Surface Area of small solid}}{\text{Surface Area of large solid}} \right) = k^2
\][/tex]
4. Calculate the surface area of the smaller solid:
- Given that the surface area of the larger solid is [tex]\(856 \, \text{m}^2\)[/tex]:
[tex]\[
\left( \frac{\text{Surface Area of small solid}}{856} \right) = (0.5)^2
\][/tex]
- Calculating [tex]\( (0.5)^2 \)[/tex], we get:
[tex]\[
(0.5)^2 = 0.25
\][/tex]
- Thus:
[tex]\[
\text{Surface Area of small solid} = 0.25 \times 856 \, \text{m}^2
\][/tex]
- Performing the multiplication:
[tex]\[
0.25 \times 856 = 214 \, \text{m}^2
\][/tex]
Therefore, the surface area of the smaller solid is:
[tex]\[
\boxed{214 \, \text{m}^2}
\][/tex]
