Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a c...

Question

Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. In a sculpture competition, precipitants are given a cylindrical wooden block and must carve a notch by making two flat cuts. The cuts extend inward until they reach the cylinder's axis and intersect along a diameter D of the circular cross section. The angle between the two cuts is fixed at 120 degrees. At what angle should the cuts be made relative to the horizontal plane that passes through D to minimize the volume of the notch. Awesome. So it appears for this particular problem, our angle, our final answer we're ultimately trying to solve words we're trying to figure out. What exact angle or what angle should these cuts be made relative to this horizontal plane that passes through the diameter D in order to minimize the volume of the notch? So we're ultimately trying to figure out what this angle value is, and that's our final answer that we're ultimately trying to solve for. So with that said, let's quickly look at the figure that is provided to us by the prom itself. So as you can see, in brown, we have our cylindrical wooden block that has our notch, that has our angle between the two cuts being 120 degrees, and we have a dotted green line going and basically it's cutting our cylindrical wooden block. It's cutting this wooden block in half, so it's like in the center. And it has a dotted line going passing through the center of our cylindrical rod. And then if we look inside of our notch where you can see it's a lighter shade of brown and it looks like a chunk has been taken out of it, we have a brown line denoting our diameter D. So now that we have a visualization of this cylindrical wooden block, let's take a moment to read off our multiple choice answers to see what our final answer pair might be, noting that they're all going to be in the same units of degrees. So A is 15 and 105, B is 30 and 90, C is 45 and 75, and D is 60 and 60. Awesome. So first off, in order for us to effectively solve for this problem, we need to investigate and understand the geometry, this geometry that is associated with this specific notch. So we need to note that the notch is created by two planar cuts that intersect along the diameter of the circular cross section and form an angle. Theta, which is equal to 120 degrees between them, and the two planes will make angles alpha and. 120 degrees minus alpha with the horizontal. So ultimately we want to find the optimal angle alpha, which this is the angle of the cuts with respect to the horizontal, that minimizes the volume of the notch. So we need to note that the notch will consist of two triangular wedges, and each wedge will have a triangular cross section where the base. R is going to be half the diameter, so it's going to be D divided by 2, and we also need to note that since tangent. Of alpha is equal to the height H divided by. R, we also need to note that H, the height is going to be equal to R multiplied by tangent of alpha. Fantastic. And in case this is Hard to understand. You're like, wait, how are we getting all of this information? Well, if it helps, you need to take our figure that's provided to us, and let's create a more detailed diagram to better visualize the inner workings. Of the geometry of this notch. So with that said, if we were to create a more detailed diagram, it should look something along the lines of this. As you can see, when you look at our notch, it will form a right triangle, and as you can see, we have our two angles. We have alpha, and then 120 degrees minus alpha. And when we split them into two separate triangles, we'll have our height H1. And then our base R, which we know that R is half the diameter, and it's gonna have some angle alpha, whereas for the height of our second triangle, which is H2, we'll also have a base R and have an inner angle of 120 degrees minus. Alpha. But if you were to line up, both of the bases are, so they're touching one another, that would mean that the height H2 is on top, and then the height H1's on the bottom, and then we'll have our two angles, alpha on the bottom and then 120 degrees minus alpha on top. So, as you can see, you can envision this notch as triangles, and we're gonna be using these triangles to help us solve for this particular problem. So with that said, we now need to figure out the area of each triangle, and the area of each triangle will be, let's start with A1 first. So the area of our first triangle, A1, is going to be equal to 1/2, and then we need to take 1/2, and we're going to multiply it by R. Then we need to multiply this all by R multiplied by tangent. Of alpha, and this will be all equal to 1/2 multiplied by R2 multiplied by tangent of alpha. And the second area or A2, it's going to be equal to 12 multiplied by R, all multiplied by R multiplied by tangent of. However, this is slightly different because it's going to be 120 degrees minus alpha. So I ran out of room a little bit, but when you simplify, similar to A1, we'll get 12 multiplied by R2d, multiplied by tangent of. 120 degrees minus alpha, when it's simplified. So now we need to determine the volume of the notch, the volume of the notch, which we'll call VN. So the volume of the notch VN is going to be equal to L multiplied by A1 plus A2, where L is going to be the length of the notch. So when we plug in our values of A1 and A2 into our expression, we will find that VN is equal to L, all multiplied by 1/2 multiplied by R squared multiplied by tangent of alpha plus 1/2 multiplied by. R squared multiplied by tangent of 120 degrees minus alpha. Fantastic. So, when we do a little bit of simplifying, you'll find that VN, the volume of a notch, is going to be equal to 1/2 multiplied by L, multiplied by R squared, multiplied by, or rather all multiplied by tangent of alpha plus tangent of 120 degrees minus alpha. Fantastic. So in order to minimize the volume of the notch, we will need to differentiate V of alpha with respect to alpha and set it equal to 0. So we need to recall that D by D alpha of Tangent of alpha is going to be equal to C2 alpha. That's what we should be able to recall. And once we recall that, we can go ahead and take the first derivative, so we could say that V of alpha, where the prime symbol means the first derivative. So V of alpha is equal to 1/2, and we need to take 1/2, we need to multiply it by L, multiplied by R. Squared, and then we need to multiply this all by drum roll, C can squared of alpha. Plus C can squad of 120 degrees minus alpha multiplied by -1. So when we simplify one more time, we'll get that V of alpha is equal to 12 multiplied by L multiplied by R2 all multiplied by C2d of alpha minus C2 of 120 minus alpha. Fantastic. So now we need to set Our V of alpha equal to 0. So when we do that, we will get 1/2 multiplied by L multiplied by R2 all multiplied by C2 alpha minus C2 of 120 degrees minus alpha. Fantastic. So now, and don't forget we need to set it all equal to 0. So now what we're doing at this point is we're going to be simplifying this expression to rearrange it to isolate and ultimately solve for alpha all by itself. So when we do that, we will find, and I'm going to skip quite a few steps in an effort to save time, but when you rearrange this expression to isolate and solve for alpha, you'll find that alpha is simply equal to 60 degrees. And This is potentially our final answer, but we cannot go ahead and say this is our final answer until we confirm that if alpha equals 60 degrees corresponds to a minimum using the first derivative test. So we need to confirm that this value alpha equals 60 degrees corresponds to a minimum by performing the first derivative test. So we need to note Just in case we don't recall that L multiplied by R2 is equal to a positive constant value, which you'll call C+. So L multiplied by R2 is a positive constant value. So, therefore, we need to note that for alpha is less than 60 degrees, let us state that So let us state that alpha is going to be equal to 59 degrees, for example. So if we plug in a value of 59 degrees in or alpha into our expression, we will find that V of 59 degrees, it's going to be equal to 1/2 multiplied by L, multiplied by R2d, all multiplied by C2d of. 59 degrees minus C squad of 120 degrees minus 59 degrees. Fantastic. So, when we plug that into our calculator, we will find that V of 59 degrees is going to be approximately equal to -0.24. 238. And then don't forget that this is going to be multiplied by L and multiplied by R squad. However, we also need to consider the condition where alpha is greater than 60 degrees. So let us say that in this case, that when alpha is greater than 60 degrees, let us say that alpha is equal to 61 degrees. So doing a similar methodology like we did for our V of 59 degrees, but instead of 59 degrees, let's plug in a value of 61 degrees. And instead of rewriting the equation out again, I will stay, just give you the answer. So V of 61 degrees is going to be approximately equal to 0.24238. Multiplied by L, multiplied by R squad. So, unlike V 59 degrees, which gave us a negative result, V 61 degrees gave us a positive result, and since the sign of the derivative changes from negative to positive, that will mean that alpha equals 60 degrees corresponds to a minimum value. So that's great. Now we need to find the other angle. So to find the other angle, we need to take 120 degrees minus our value for alpha, which we determined alpha to be 60 degrees, so 120 degrees minus 60 degrees gives us 60 degrees as our final answer. And that's it. We solve for this problem. Hooray. So looking at our multiple choice answers, the correct answer, without a doubt, will have to be Multiple choice answer letter D, which once again is 60 degrees and 60 degrees. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Volume of a Solid of Revolution

Optimization in Calculus

Geometric Interpretation of Angles

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