Volume of a torus The volume of a torus (doughnut or bagel) wi...

Question

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.

a. Find db/da for a torus with a volume of 64π².

Accepted Answer

Welcome back, everyone. The surface area of a cone is given by the formula S E equals pi R multiplied by R plus the square root of R2 + H2, where R is the radius and H is the height of the cone. Calculate the value of the derivative of R with respect to H if S equals 1400 pi. A says it's negative R H divided by 2 r multiplied by the square root of R2 + H2 + 2 r2 + H2. B says its negative R H divided by 2 R multiplied by the square root of R2 + H2 plus R2 + 2 H2. See, negative RH divided by the square of R2 + H2 + 2 R2 + H2 and D R H divided by 2R root R2 + H2 + 2 R2 + H2. Now, we're trying to find a value if S equals 1400 pi. That means we can substitute 1400 pi into our formula. And when we do that, so substituting S equals 1400 pi. OK. Then we're going to get pi R multiplied by R plus the square root of R2 + H2 to be equal to 1400 pi. Now, if we're trying to calculate the value of the derivative of R with respect to H, then we'll need to differentiate differentiate uh our expression R with respect to H. But notice that it's not purely in terms of R, so we'll need to differentiate implicitly, OK? So differentiating implicitly. OK, with respect to age. OK. Then notice that on our left hand side we can differentiate implicitly using the product rule because here we have our product i R multiplied by R plus the square root of R2 + H2. And by applying the product rule, then we would get i multiplied by R + square of R2 + H2. Multiplied by DR with the derivative of R with respect to H plus pi R multiplied by D R D H plus 2 R D R D H plus 2 H read that properly here all divided by 2 multiplied by the square root of R2 plus 82, OK. So this is what we'd get on our left hand side and on our right hand side, the derivative of 1400 pi with respect to H is 0. Now notice here that we can cancel through our equation for pi. And no. If we were to expand here our brackets, then we would get R D R D H plus the square root of R squad plus 8 squared. OK, multiplied by DRDH. OK, plus RDRD H plus 2 RDRDH plus 2 H divided by 2 square root of R2 plus 82, OK. Uh, multiplied by, well, multiplied by R here. So if we multiply through by R, then it's gonna be 2 R squared. So let me just write it down here instead of writing it beside it. It's gonna be 2 R squared here, OK, and 2 are H. We've distributed the R across our terms. I remember now all of that is going to be 0. Now how could we simplify this anymore? Can we, can we make this any simpler here? Well, if we multiply through our entire equation by, let me write this properly, if we multiply through by the square root, or sorry, the denominator here 2 multiplied by the square root of R2 + 82, then you should find that we'll end up with 4 R multiplied by the square root of R2 + 82. The IDH. OK. Plus 2 multiplied by R2 + H2 DRDH, OK, plus 2 R 2 DRDH. Plus 2 R H to be equal to 0. So we've basically multiplied this entire expression by 2 root R2 + H2 to get it out of the denominator. Now, we can start to group any term that has DRDH because that's that's what we're trying to find the derivative of R with respect to age, OK? Now notice here, OK, that if we do that. We go ahead and we do that here. We can expand the 2 multiplied by R2 + 82, and then we're gonna get 4 R multiplied by the square of R2 + 82, OK, plus 2 R2 + 2R2, which is 4 R2, plus 2 H2, OK? all multiplied by DRDH, OK. And no if we add 2 RH to both sides, then this is gonna be equal to negative to RH. Now all we need to do now is to solve for DRDH. First of all, OK? Notice here that we can factor 2 from both sides, OK? 2 into 4R goes 2 times, 2 into 4 are squared also goes 2 times 2 into 2H. So into 2 here goes 1 time and 2 into 2 goes 1, OK. And know if we divide through by our entire expression here, then we should get DRDH OK, to be equal to negative RH all divided. By 2 are multiplied by the square of R2 plus 82, the term that would be here plus 2 are squared. OK, the term that would be here plus H squared or final term. Thus, the derivative of R with respect to H equals negative R H divided by 2 R root R2 + H2 + 2R2 + H2 if S equals 1400 pi. Therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
a. Find db/da for a torus with a volume of 64π².

Key Concepts

Volume of a Torus

Differentiation

Implicit Differentiation

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