Right circular cone The lateral surface area ...

Question

Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation

______

S = πr √ r² + h².

c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Accepted Answer

Hello there. Today we're going to 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. A cylindrical water tank is expanding due to the increasing water pressure. Both the radius of the base and the height of the cylinder are changing over time. The lateral surface area of the cylinder is given by the formula. S is equal to 2 multiplied by i multiplied by R multiplied by H. If the radius and the height of the tank vary with time, how can we express the rate of change of the lateral surface area, DS by DT in terms of the rates of change of the radius DR by DT and the height DH by DT? Fantastic. So for for this particular problem it appears the final answer that we're ultimately trying to solve for is we're trying to figure out a way to express the rate of change of the lateral surface area, which is DS by DT, written in terms of the rates of change of the radius, DR by DT, and the height DH by DT. So now that we know that we're ultimately trying to create some sort of expression that describes the rate of change of the lateral surface area, DS. DT in terms of DR by DT and DH by DT. Our first step that we need to take in order to solve for this particular prompt is we need to differentiate both sides of the given formula for the lateral surface area, meaning we need to take S equals 2 multiplied by pi, multiplied by R multiplied by H, and we need to differentiate this equation with respect to time T. And in order to do that, we need to recall and apply the product rule for differentiation, since R and H are both functions of T. So as we should recall, the product rule states that D by DT is, so we need to take D by DT of U multiplied by V, which is going to be equal to V multiplied by U plus U multiplied by V. So that means we need to take D by DT of S is equal to D by DT of 2 multiplied by i multiplied by R multiplied by H. Fantastic. So now, Our next step is we need to take DS by DT is equal to 2 multiplied by H multiplied by D by DT of R plus R multiplied by D by DT of H. Fantastic. So that will mean when we simplify that DS by DT, so DS by DT, which is our final answer we're ultimately trying to solve for, so we can say that DS by DT and noting that this S is a capital S, so DS by DT is going to be equal to 2 pi multiplied by H multiplied by DR by DT. Plus are multiplied by DH by DT. And that's it. That is our final answer. Hooray, we did it. So, therefore, without a doubt, our final answer has to be DS by DT is equal to 2 pi multiplied by H, multiplied by DR by DT plus R multiplied by DH by DT. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation
______
S = πr √ r² + h².

c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Key Concepts

Related Rates

Differentiation

Chain Rule

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Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation ______S = πr √ r² + h².c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?