I want to do a quick correction to the AIME problem where we cut out a wedge of a log in that problem. So we had a wedge so just to remind ourselves that we had a log and it looked like that we made one cut that was right along that was perpendicular to the axis of the log and we made another cut that had a 45 degree angle. So it was like that. And the solution to the problem we had to figure out the volume of the wedge and the big trick there was if I drew the wedge like this so let me take a side view of the wedge. So this is the side of the wedge right over here. I could shade it to show that it's kind of round. So if I were to shade it you could kind of see that it is round. So this is a side view of the wedge. We knew that this right here is a 45 degree angle. And the way that we solved the problem or the kind of trick was to say Hey, if we had another wedge like this and we just stacked it on top of this one if we just flipped it over and made it like this then we also have a 45 degree angle over here and it would make a cylinder and it's very easy to figure out the height of a cylinder. And I misspoke when I said it was lucky that we had a 45 degree angle. Because in the last video I was actually visualizing it incorrectly. Because I said if you had anything less than a 45 degree angle this wouldn't have worked out properly But I was wrong! It would have worked out properly. Let me show you. So this, let's just say whatever angle you have here. So this is going to be 90 degrees over here. Whatever angle you have over here so let's call that theta this over here, they have to add up to 180. So theta plus 90 plus this has to be 180 Or theta plus this has to be 90 or you could just call this 90 minus theta. Now, if I were to take the same thing if I took another kind of wedge like this and flipped it over this thing it's not this angle that's going to be right here it's this angle. Let me just draw it over here. It would look like this. Where this now is theta. This end right here is actually that end on the flipped version. And now this angle right over here is now 90 minus theta. So clearly you have 90 minus theta plus theta. So this thing is going to be a right angle no matter what. Obviously it has to be a reasonable angle but no matter assuming that its less than 90 degrees it would have worked out. You could have done the same trick. Anyway, hopefully that clarifies it and I apologize for the incorrect visualization in the last video.