Find the rate of change for volume of a cylinder with respect to radius if the height is equal to the radius.
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Find the rate of change for volume of a cylinder with respect to radius if the height is equal to the radius.

[From: ] [author: ] [Date: 11-10-04] [Hit: ]
=======================-If the height of the cylinder is equal to the radius, then h=r. Then, the equation for the volume of a cylinder is V=(pi)(r^3). Now, taking the derivative of V with respect to the radius,......
This is Calculus 1 material, rates of change (derivatives) more specifically. There are no actual values, so everything is to be done with variables, short of the numbers in the equation: V= pi*(r^2)*h. I have no idea how to start this, so please show all steps and explain, thank you.

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V= pi*(r^2)*h.

It is given r = h

hence

v = πr^3

dv/dt = 3πr^2*dr/dt

dr/dt is the rate at which the radius varies.
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If the height of the cylinder is equal to the radius, then h=r. Then, the equation for the volume of a cylinder is V=(pi)(r^3). Now, taking the derivative of V with respect to the radius, the variable r, we get dV/dr=3pi(r^2)(dr/dt) by the power rule and the chain rule of differentiation. dr/dt represents the rate of change of the radius, r represents the current snapshot length of the radius as it changes and dV/dr represents how the volume changes at that moment.
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