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Sphere B is the image of sphere A after dilation by a scale factor of $\frac{1}{3}$. If the surface area of sphere $B$ is $7 \mathrm{~km}^{2}$, find the surface area of sphere $A$, the preimage.
Answer: 0.78
Answer
\(\boxed{S_A \approx 0.78 \mathrm{km}^2}\)
Step 1 ：Given the surface area of sphere B is 7 km^2 and the scale factor of dilation is \(\frac{1}{3}\).
Step 2 ：Set up a proportion between the surface areas of sphere A and sphere B: \(\frac{S_A}{S_B} = \left(\frac{1}{3}\right)^2\).
Step 3 ：Plug in the given value of $S_B$: \(\frac{S_A}{7} = \left(\frac{1}{3}\right)^2\).
Step 4 ：Solve for $S_A$: \(S_A = 7 \cdot \left(\frac{1}{3}\right)^2\).
Step 5 ：Calculate the surface area of sphere A: \(S_A \approx 0.78 \mathrm{km}^2\).
Step 6 ：\(\boxed{S_A \approx 0.78 \mathrm{km}^2}\)
