Right circular cylinder The total surface are...

Question

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.

b. How is dS/dt related to dh/dt if r is constant?

Accepted Answer

Welcome back, everyone, to another video. The total surface area of the right circular cone, including its bases given by S equals by R multiplied by R + square root of R2 + H2 where R is the base radius and H is the height. Determine how DS divided by DT is related to DH divided by DT if R is constant. So essentially we want to find the relationship between DS divided by the T and DH divided by DT. We have to understand that S is a function of height and height is a function of time. So we're dealing with a composite function, and if we want to identify the derivative of DS divided by DT, as a result, we have to apply the chain rule, which means that first of all, we're differentiating S. With respect to H and then multiplying it by the derivative of H with respect to T. So this gives us our relationship, and this means that we have to identify the derivative DS divided by DH. So let's go ahead and identify that derivative. So we want to identify S with respect to H, which is the derivative of pi R. Multiplied by R plus square root of R2 + H2. We know that R is a constant, so we can factor out i R. This gives us pi R multiplied by. The derivative of R plus the derivative of R2 + H2 raise the power of 1/2. So we're going to rewrite it in its exponential form, right? So let's valuate the results. We're going to get pi R multiplied by the derivative of R is 0 because radius is a constant, and this leaves us with one term to differentiate. It is again a composite function. So first of all, we are applying the power rule which gives us 1/2 multiplied by R2 + H2, raise the power of -12. So according to the power rule, we are bringing down the exponent and decreasing the original exponent by one unit. And now according to the chain rule, we're multiplying by the derivative of the inner function, which is the derivative of R2 plus H2. Let's simplify. This gives us pi R multiplied by. Now we can rewrite one half multiplied by R2 + H2 raise the power of -1/2 as a fraction. So in the denominator we have 2 and R2 + H2 raise the power of 12, right, because we originally had a negative exponent and 12 represents square roots. So we can say 2 square root of R2 + H2. Is our denominator. In the numerator, we can add our derivative, the derivative of R20 because R is a constant and the derivative of H2 is 2 H. Based on the power rule. And we can see that we can cancel out the twos. Which gives us I R H in the numerator. And square root of R squared plus H squared in the denominator. Well then, so we have identified the factor that we need for the relationship, and this gives us DS divided by DT. Is equal to IRH. Divided by a square root of R2 + H2 multiplied by the H divided by the C. So that is our final relationship. Let's label it as our final answer, and thank you for watching.

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.

b. How is dS/dt related to dh/dt if r is constant?

Key Concepts

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Differentiation

Implicit Differentiation

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Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.b. How is dS/dt related to dh/dt if r is constant?