How long will it take Micrometeorites to cover the moon ?

1. The problem statement, all variables and given/known data
If one micrometeorite (a sphere with a diameter of 1.50×10-6 m) strikes each square meter of the surface of the Moon each second (1 micrometeorite/m2/s), how many years will it take to cover the Moon to a depth of 1.80 m?
Assume that after impact, the micrometeorites spread evenly over the surface of the moon.


2. Relevant equations
V = 4/3πr^3




3. The attempt at a solution
I'm not sure if I'm thinking through this correctly, but here it goes.

SA of Moon (SA) = 3.8x10^13 m^2 (got this out of my textbook)
Thickness (T) = 1.8 m (Given in problem statement)
Total Volume needed to be filled by Micrometeorites = SA * T
= (3.8x10^13)(1.8) = 6.84x10^13 m^3

Volume of 1 Micrometeorite = V = 4/3πr^3 where r = (1.5x10^-6 / 2)
= 4/3π(1.5x10^-6 / 2)^3 = 1.77x10^18

Micrometeorites needed = Volume needed/Volume of Micrometeorites
= (6.84x10^13)/(1.767x10^-18) = 3.87x10^31

Since the rate at which they fall is 1 Micrometeorite/s that means a total of
3.87x10^31 seconds

which comes out to 1.22 years.


Apparently this is wrong (we have to enter our answers online, and we're told if we're right or not). Please help!

 

True about the years - I really messed that up. How am I supposed to calculate the spherical shell?

 

I keep trying but it's not working out. Could you give me a little more?