The moon's inhabitants use a unit of distance called lunar. The number of square lunars in the moon's surface

Question

The moon's inhabitants use a unit of distance called lunar. The number of square lunars in the moon's surface is the same as the number of cubic lunars in the moon's volume. Assume the moon's diameter to be 2160 miles. How long is a lunar in miles?

The moon's inhabitants use a unit of distance called lunar. The number of square lunars in the moon's surface is the same as the number of cubic lunars in the moon's volume. Assume the moon's diameter to be 2160 miles. How long is a lunar in miles? Please explain how to get this answer....

Answer 1

360 miles.

The volume of a sphere is 4πr^3 / 3
The surface area of a sphere is 4πr^2

When r is measured in lunars, we have:
4πr^3 / 3 = 4πr^2
Divide both sides by 4πr^2
r/3 = 1
r = 3 lunars.

3 lunars = 2160/2 = 1080 miles
1 lunar = 1080/3 = 360 miles

Answer 2

You can show that for distances in miles that:
6/2160 = surface area/volume for the moon.
In general,
6 D^2/ D^3 = SA/V
If the lunar is such that SA/V= 1, Then 6 D^2 = D^3. For this to happen, the lunar is D/6 or 360 miles.