2003 AIME II problem 5
"A cylindrical log has a diameter of 12 inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a 45 degree angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as n pi, where n is a positive integer. Find n." So let's think about it. So let's just draw what they're describing. So we have a cylindrical log that has a diameter of 12 inches. So let's draw that. So let me draw it like this. So that is a cross section of the log. It has a diameter of 12 inches. So this is 12 inches. And it is a cylinder, so it looks like this. That is our log. That is the log in question. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first cut is perpendicular to the axis of the cylinder. So it would go-- it'd really just cut it-- it would just cut it straight. That's another way to think about it. It's perpendicular to the axis of the cylinder. The axis of the cylinder goes through the cylinder, like that. I don't want to do that; it'll make the diagram a little confusing. If the log was transparent, our cut would look like this. This first cut would just go through the log like this. It would really just form a cross section of the log. So that's the first cut. The first is perpendicular to the axis of the cylinder. And the plane of the second cut forms a 45 degree angle with the plane of the first cut. So that thing is going to come in at a 45 degree angle. So it's going to cut into our log something like that. Something like that, right over there. The intersection of these two planes has exactly one point in common with the log. That would be this point, right over here. So we need to find the number of cubic inches in the wedge, expressed as n pi, and then we have to figure out n. So let's draw this wedge, this wedge that we've cut out here. So this right here is our wedge. Let me take it out and kind of flip it around. So the base of-- I'll make the base of the wedge the perpendicular cut. This cut right over here. So that is the base of the wedge. That is our base. And then you can do the top of our cut to be the magenta, the 45 degree angle cut. So maybe I'll draw it like this. So it would look something like this. I'm trying my best to draw it. That's really the hard part, here. So let me draw the 45 degree angle cut. It's going to look something like that, where this angle right over here-- so if I were to go-- the diameter of this top thing, it's actually, it's not going to be a normal circle. It's going to be more elliptical. But the diameter of this top thing versus the diameter of the base, which is the diameter of a circle, this right here is going to be a 45 degree angle. Now when I first looked at this problem, there's all sorts of temptations here. Maybe you use some calculus. Maybe you use some-- you rotate something around some axis to find the volume. Maybe you can take some type of average, here. And you probably could do something like that. But the easiest thing here-- and this is kind of-- whenever you see these kind of competition-type math problems-- and this comes from the 2003 AIM exam-- is that there should be a quick way to do it. And for this exam, particular, you shouldn't have to use any type of calculus. And so, if you find yourself doing something tedious and hairy, you're probably not seeing the easiest way to do this problem. And this problem is actually ridiculously easy to do, if you just see the trick here. And the trick here is, instead of just directly trying to solve for the volume of this figure, right here, take another one of these guys, and flip it over, and put it on top of him. So let's say, if you did that, you would have another thing on top of it, just like that. So if I just took two of these wedges and stacked them on top of each other-- flipped one of them and stacked them on top of each other, it would look like that. This is another wedge right here. So I took their angled faces and put them right on top of each other. And so if you took two wedges together, flip one of them, and put it on top of the other, what do you get? So the equivalent-- I could draw the green wedge right over here. The green wedge would look like this, where its base looks like this. So what would two wedges put in this configuration look like? Well, it's just a cylinder, now. And it now is a cylinder. So this is a cylinder with a diameter of 12. So that diameter is 12. But in order to figure out the volume of a cylinder, you still need to figure out the height of the cylinder. You still have to figure out this length, right over here. You still have to figure out what the height is equal to. You have to figure out what this length, right over here, is equal to. And that's where the 45 degrees helps us. Well, the 45 degrees already helped us, because if you flipped the thing over and put it on top, it forms a nice cylinder for us. If it was another angle, it wouldn't have been a nice, clean cylinder, like this. But the 45 degrees also tells us what this height is. And let's just think about it, here, for a second. This triangle that I had drawn, in the beginning-- let me draw this in-- well, I already used blue. Let me draw it in yellow. So if I were to take the diameter of the angled surface-- and once again, it's not a circle, but it's kind of a stretched-out circle-- and I took the diameter of the base, they form a 45 degree angle. So this is 45 degrees. This right here is also going to be 45 degrees. This length, over here, we know-- we know this length, right over here, is going to be 12. Now, this right here is a 45-45-90 triangle. Let me draw it like this. I could draw it like this. This right here is 45-45-90 triangle. And you might say, how did I know that's 45? Well, this is going to be a right angle, right over here. And the sum of the angles have to add up to 180. And then, if you have 45 and 90 already, then this one has to be 45 degrees. And in 45-45-90 triangle, this side is equal to that side. It is an isosceles triangle. The two base angles are the same, so the two sides are going to be the same. So if this side, right over here, is 12, then this side, right over here-- this side, right over here is 12. So the height is going to be equal to 12. So let's figure out the volume of this cylinder that's essentially two wedges, and then we can take half of that to find the volume of one wedge. And to find the volume of a cylinder, you just have to find the area of the top of the cylinder. And to find that, it's going to be pi times the radius squared. The radius here is 6, 1/2 of 12. So the area is pi r squared. So it's 36 pi-- that's that area-- times the height-- times 12. And so the whole volume, our volume is going to be equal to-- what's that? 360 plus 70. Let me just multiply it out. I don't want to make a careless mistake. 36 times 12. 36 times 2 is 72. 1 times 36 is 36. 2, 13, 4. So it's 432 pi. Now we have to be very careful. This is the volume of two wedges. So this is the volume of two wedges, I could call it. So the volume of one wedge, is going to be 1/2 of this. Let me do this in a different color. The volume of one wedge is going to be 1/2 of this, or 216 pi. And so if we want to find n, because they say, the number of cubic inches in the wedge can be expressed as n pi. It's 216 pi, where n is a positive integer. Find n. Well, we just figured that out. n is 216.