The Wednesday Morning Math Challenge: Week 10 Answers

Last week, we tackled a thought experiment about how many stars are behind the moon (if you haven’t yet, you can check out the problem by clicking here ). Our answer won’t be the same one that an astrophysicist would come up with. But by simplifying the question and using some rough assumptions, we’ll get a crude estimate.

We are given four assumptions:
1. There are 100 billion stars in the Milky Way.
2. The stars in the Milky Way are evenly distributed around the earth.
3. The moon is 385,000 kilometers away from the earth.
4. The moon has a radius of 1,700 kilometers.

Since we know that the stars are evenly distributed around the earth, it suffices to calculate the proportion of the sky covered up by the moon, and then multiply that by the number of stars (which we are told is 100,000,000,000). So how do we figure out the proportion of the sky covered up by the moon?

We can think of the earth as nestled at the center of a sphere with a radius that is equal to the distance of the moon to the earth (or 385,000), and then calculate the surface area of that sphere. Since the surface area of a sphere is equal to 4 x pi x r^2, we have:

4 x pi x (385,000)^2 = 1862650300000

(Note, of course, that I’m rounding. We are after a rough approximation.)

Now we need to calculate the area of the moon to figure out what proportion of the sphere it covers. There’s a wrinkle here. To do the problem rigorously, we would need to consider the moon not as a circle on top of a plane, but as a spherical cap set against a sphere. How would we go about doing that? And in what circumstances is our approximation here valid? I encourage you to think about this.

Since we are using such rough numbers, however, I am going to simplify the problem and treat the moon as a circle. We should still get a very rough approximation. Since the area of a circle is pi x r^2, we can calculate the area of a cross section of the moon to be:

pi x 1,700^2 = 9079203

So to figure out the proportion of the sky that the moon covers, we simply divide the area of the moon by the area of the sky.

9079203 / 1862650300000 = 0.00000487434

Now we can multiply that by the number of stars in the Milky Way, which gives us 487434.

So there are roughly 490,000 stars in our galaxy hidden by the moon!