The formula for the surface area $A$ of a sphere is given by:

$A = 4\pi r^2$

where $r$ is the radius of the sphere. We are given that the surface area is $3600\pi$ square miles. So, we can set up the equation:

$4\pi r^2 = 3600\pi$

Now, divide both sides by $\pi$:

$4r^2 = 3600$

Next, divide both sides by 4:

$r^2 = 900$

Take the square root of both sides to find $r$:

$r = 30 \text{ miles}$

Now that we know the radius, we can find the volume $V$ of the sphere using the formula:

$V = \frac{4}{3} \pi r^3$

Substitute $r = 30$ miles:

$V = \frac{4}{3} \pi (30)^3$

First, calculate $30^3$:

$30^3 = 27000$

Now substitute that into the volume formula:

$V = \frac{4}{3} \pi (27000) = \frac{4}{3} \times 27000 \times \pi = 36000 \pi \text{ cubic miles}$

Thus, the volume of the moon is:

$V = 36000\pi \text{ cubic miles}$

If you want the volume in decimal form, it is approximately:

$V \approx 113097.34 \text{ cubic miles}$

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. How does the volume of a sphere change if its radius doubles?
2. What is the relationship between the surface area and volume of a sphere?
3. How can we calculate the surface area if we are given only the volume?
4. What is the volume of a sphere with a surface area of $144 \pi$ square miles?
5. How does the formula for a sphere's volume compare with that of a cylinder?

Tip: When scaling dimensions in geometry (like doubling the radius of a sphere), note that surface area scales with the square of the radius, while volume scales with the cube of the radius!