 Here's how to solve for the path length of a P-wave through the mantle on the assumption that the mantle is homogenous and transmits waves at the same speed regardless of depth. So I'm just going to draw a hemisphere of the Earth, hopefully. And here's the center of the Earth. Now, let's assume that there's an earthquake that happens over here, and you've got a seismometer over here that records a P-wave. So if the mantle is homogenous, then it means that the P-wave will travel a straight line path through the Earth, from the earthquake to the seismometer, like this. Now, to solve this problem of what this distance X is, I'm going to first draw a radius from the center of the Earth to my earthquake, and another radius from the center of my Earth to the seismometer. This is an isosceles triangle because these two sides are both the radius of the Earth, and then this other side here is the one we're trying to find. Now, we also know something else about this triangle. We know this angle. This angle, a seismologist called Delta, and it is exactly the great circle arc distance between the earthquake and the seismometer. So you've already calculated Delta for all the stations that measured this earthquake in this problem set. Hooray! Now, if we drop a perpendicular line from X to the center of the Earth, so bisecting X and bisecting this angle, Delta, that would look like this. Okay, now this is a right angle, and this distance here is one-half X, and then this angle here is one-half Delta. Using the property of right triangles, we can see that the sine of one-half Delta is equal to the length of the opposite side, which is one-half X, divided by the hypotenuse, which is the radius of the Earth. So let's write that down. The sine of one-half Delta equals one-half X over the radius. And this is a fabulous expression because we know everything in it except X, and we're trying to solve for X. So that's awesome. Let's rearrange. So X equals twice the radius times the sine of one-half Delta. So if you want to find out X, you can just plug in Delta, which you've already calculated, and the radius of the Earth, which we can take as 6,371 kilometers, that's a good approximation, and away you go. That X is the path length that a p-wave takes through the mantle if you assume that the mantle is homogenous.