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: $\mathrm{km}^{2}$ Submit Answer

Step 1 ：Given that sphere B is a dilation of sphere A by a scale factor of \(\frac{1}{3}\) and the surface area of sphere B is 7 km^2.

Step 2 ：Using the relationship between the surface areas of similar figures: \(\frac{\text{Area of A}}{\text{Area of B}} = (\text{Scale factor})^2\)

Step 3 ：Substitute the given values: \(\frac{\text{Area of A}}{7} = \left(\frac{1}{3}\right)^2\)

Step 4 ：Solve for the Area of A: \(\text{Area of A} = 7 \times \left(\frac{1}{3}\right)^2\)

Step 5 ：Calculate the Area of A: \(\text{Area of A} \approx 0.78\)

Step 6 ：\(\boxed{\text{The surface area of sphere A is approximately 0.78 km}^2}\)