The total surface area S of a right circular cylinder is relat...

Question

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.

a. How is dS/dt related to dr/dt if h is constant?

Accepted Answer

Welcome back, everyone. A right cone's lateral surface area L is given by the equation L equals by R square root of H2 plus R2, where R is the base radius and H is the height. If R is constant, how is D L divided by DT related to D H divided by DT? For this problem, let's understand that L is a function of height H because R is constant and H is changing, but H is a function of time, right? So we have a function L of H of T, which is a composite function. And this means that if we want to evaluate the derivative D L divided by the T, right, the derivative of L with respect to the independent variable T, we have to apply the chain rule which says that first of all we are differentiating L. With respect to heights, we are taking DL divided by DH and then according to the chain rule, we're multiplying that by the derivative of H with respect to T, so we're multiplying by DH divided by DT. And as we can see, we have the relationship between DL divided by DT and D H divided by Dt in terms of the multiple DL divided by DH. This means that we want to evaluate the derivative D L divided by DH. How can we do that? Well, we're going to differentiate pi R multiplied by. H2 + R2 raises the power of 1/2. So basically what we're doing is using the original expression of L. we are factoring out the constant because R is radius, and we are rewriting our radical in terms of the exponent, right? This gives us pi R, which is our constant. Now we have a composite function, so first of all, we are applying the power rule. We get 1/2 multiplied by H2 + R2 raise to the power of -12 according to the power rule, and because we have an inner function according to the chain rule, we also have to multiply by the derivative of H2 plus R2. So now let's keep calculating the result. This gives us pi R multiplied by 1/2 multiplied by H2 + R2, raise the power of -12 multiplied by 2H because the derivative of H2 is 2H according to the power rule, and the derivative of R2 is 0. It is a constant. And now we can see that 1/2 multiplied by 2 is 1. And we also have H2 plus R2 raise the power of -12 so we can use it in our denominator as a square root, right, because we have a negative exponent. So we're going to have square root of H2 + R squad in the denominator, and then in the numerator we simply have pi R multiplied by H. So we have obtained the required multiple that expresses the needed relationship. This means that the relationship can be written as dl divided by D T equals our derivative by R H divided by square root of. H2 plus R2 multiplied by DH divided by DC. Well done, we can label our final answer and thank you for watching.

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.

a. How is dS/dt related to dr/dt if h is constant?

Key Concepts

Related Rates

Differentiation

Chain Rule

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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.a. How is dS/dt related to dr/dt if h is constant?