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Derivative of the volume of a cone, given numbers

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I solved the following problem myself:

"The radius r(t) of a cone is increasing at a rate of 3 centimeters per second and the height h(t) of the cone is decreasing at a rate of 4 centimeters per second. At a certain instant t_0, the radius is 8 centimeters and the height is 10 centimeters. What is the rate of change of the volume V(t) of the cone at that instant?"

and have come up with the answer -64π. Can someone verify this?

May 6, 2022

#1
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Here is my attempt:

Since: $$V_{cone}=\frac{1}{3}\pi r^2h$$              (*)

and given: $$\dfrac{dr}{dt}=3$$, and $$\dfrac{dh}{dt}=-4$$

Differentiate (*) with respect to t

$$\dfrac{d}{dt}V(t)=\frac{1}{3}\pi(2r\frac{dr}{dt}h+r^2\frac{dh}{dt})$$

We want to find the rate at $$t_0$$ and we are given r=8 and h=10:

$$V'(t)=\frac{1}{3}\pi(2(8)(3)+(8)^2(-4))=\frac{224}{3}\pi$$

May 6, 2022
#2
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For the differentiation part: notice that r(t) and h(t) are functions of time and they are varying. So when we differentiate r, we do the normal differentiation (r -->1) but then applying the chain rule, we multiply by r' which is dr/dt, and similarly for h). So to differentiate (*) you need to know the product rule and chain rule.

Guest May 6, 2022
#3
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I believe you meant to write $$1/3π(2(8)(3)(10)+(8)^2(-4)$$. Other than that, great work!

Guest May 6, 2022