Geometry word problem: Earth & Moon radii

The dimensions of the Earth and moon are in relationship to each other forming a golden triangle. Represented by phi, the golden ratio is the only number which has the mathematical property of its square being one more than itself. And there's a whole video on phi on the Khan Academy, and I suggest you watch it. It will give you chills. And if you watch that and you do this problem here, you'll have even more chills. But let's just try to tackle this problem. So they say that phi plus 1 is phi squared, which is neat by itself. And this is where they just write out what phi looks like. So phi is approximately 1.61803. It keeps going on and on and on. Plus 1 is going to be 1.61803 on and on and on squared, which is 2.61803. So it's just another way of expressing this. They tell us by applying the Pythagorean equation to this equation, a right triangle with sides phi square root of phi and 1 is constructed. So what are they saying? Let's say, well look, this looks kind of like the Pythagorean equation, or the Pythagorean theorem. If we make this a squared, we make this b squared, and we make this c squared, you could think about that as expressing the relationship between the sides of a right triangle where our hypotenuse c is equal to phi, that this side, the shorter side b, is equal to 1-- the square root of 1 is just 1-- and the longer side, but not the longest, the longer of the two non hypotenuse sides, is going to be the square root of phi. That's all they're saying with that first sentence. As shown below, and this is a little bit mind blowing. As shown below, the radii of the Earth and moon are in proportion to phi. So this is exciting. Let me do this in a color you can see. If you were to take the Earth's radius-- So that's this radius right over here. I keep wanting to change colors, and I'm having trouble. So if you were to take the Earth's radius and add it to the moon's radius, the sum of those two radii, the ratio of that sum to Earth's radii is the square root of phi. And that just makes you think about the universe a little bit. You should just pause that video and ponder. Who cares about this actual question? Well, we should answer the question as well, but that's kind of eerie because this isn't the only place this shows up. This shows up throughout nature and throughout mathematics. It's just a fascinating number for a whole set of reasons, and this is just a little bit eerie. But anyway, we have a problem to do. If the radius of the earth is 6,371 kilometers, then what is the radius of the moon? So let's actually redraw this triangle here, but let's draw it in terms of kilometers. Right over here, the measurements are in terms of Earth radii. This right over here is 1 Earth radii. If this is 1 earth radii, then the entire distance, the combined radii of the moon and the Earth, is square root of phi Earth radii, and the hypotenuse of this triangle is phi Earth radii. So this is in terms of Earth radii, but let's redraw the triangle, and let's draw it in terms of kilometers. So let's draw it in terms of kilometers. I'll try to draw it pretty similar to that. So if we draw it in terms of kilometers, and I'm going to do a very rough drawing of it. So that's the Earth right over here. I'll just draw a part of the Earth. I don't have to draw the whole thing. I think you get the idea. And then this is the moon right over here. I think you get the idea. They told us that the radius of the Earth is 6,371 kilometers. Well, they also told us that this height, the height of this right triangle, is square root of phi Earth radii. So if we write it in terms of kilometers it's going to be 6,371 times the square root of phi kilometers. Square root of phi Earth radii. So that's this entire distance right over here. Now what they want us to figure out is the radius of the moon. They want to figure out this distance right over here. So let's call that r for the radius of the moon. So how can we figure out r? Well, we also know what this segment is. This segment, which I will do in green, is also the radius of the Earth. The Earth is roughly a sphere. So we could say that this distance right over here is also 6,371 kilometers. So this actually breaks down to a fairly simple problem. The combined radii, we can write it in two different ways. We could write it as the radius of the moon plus the radius of the Earth, which is 6,371 kilometers. And we'll just assume that everything we're doing is in kilometers now. And we can write it as, the combined radii, as 6,371 times the square root of phi. Once again, the combined radii is square root of phi times the length of the Earth's radius. So here we just have it in terms of Earth radii. Here we have it in terms of kilometers. This is the Earth's radius. You multiply that times square root of phi, you get the combined radius. Well, now we just have to solve for r. We can subtract 6,371 from both sides, and we get r is equal to 6,371 times the square root of phi, minus 6,371. And if we want, we could factor a 6,371 from both of those terms, and we would get r is equal to 6,371 times square root of phi minus 1. And we are done. And this is still pretty neat and pretty cool.