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Current time:0:00Total duration:8:14

2003 AIME II problem 5

Video transcript

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 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 there 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 we have a so let me draw it like this so that is a cross-section of the log and has a diameter of 12 inches so this is 12 12 inches and it is a cylinder so it looks like this that is our log so 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 is perpendicular to the axis of the cylinder so it would go it would really just cut it it would just cut it straight there's another way to think about it it's perpendicular to the axis the axis of the cylinder the axis of the cylinder goes through the cylinder like that I don't want to do that and make the diagram a little confusing if the log was transparent our cut would look like this our this first cut would just go through the log like this it would really just form kind of a cross section of the log so that's the first cut the first is perpendicular to the axis of the cylinder and the 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 in it's going to cut into our log something like that something like that right over there the intersection of these two planes is 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 then we have to figure and so let's draw this wedge this wedge that we've cut out here so this is 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 kind of view 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 it would look something like this trying my best to draw it it's really the hard part here so let me draw the 45 degree angle cut it's gonna look something it's going to look something like that and where this angle right over here so if I were to go diameter of this top thing and 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 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 use some calculus maybe use some you know you rotate something around some axis to find the volume maybe you can take you know some type of average here and you probably could do something like that but the easiest thing here and this is kind of you know whenever you see these kind of competition type math problems and this comes from the 2003 aim exam is that there shouldn't be a quick way to do it and if you and and for this exam particularly 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 put it and flip it over and put it on top of him so let's 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 it would and stacked them on top of each other flipped one of them and stacked them on top of each other it would look it would look like that this is another wedge right here so either I took they're kind of angled face 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 what's just a cylinder now and it now is a cylinder it's now 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 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 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 I already used blue let me draw it in yellow so if I were to take the kind of diameter of the angled surface and once again it's not a circle but it's kind of a stretched out circle and then I take 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 just draw it like I could draw it like this this right here is 45-45-90 triangle 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 a 45-45-90 triangles this side is equal to that side is it not 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 so the volume and to find the volume of a cylinder you just have to find the area 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 six half of twelve so the area is PI R squared so it's 36 PI that's that area times the height times 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 make a careless mistake 36 times 12 36 times 2 is 72 1 times 36 is 36 213 4 so it's 400 430 2 pi now we have to be very careful this is the volume of two wedges so this is a volume of two wedges I could call it so the volume of one wedge the volume of one wedge is going to be half of this we do this in a different color the volume of one wedge is going to be half of this or 216 pi and so if we want to find n since they say the number of cubic inches in the wedge can be expressed as n pi is 216 pi where n is a positive integer find n well we just figured that out and is 216