Artificial intelligent assistant

Rate of change for a volume I have the following question : _The radius of a right circular cone is increasing at a rate of 5 inches per second and its height is decreasing at a rate of 4 inches per second. At what rate is the volume of the cone changing when the radius is 30 inches and the height is 20 inches?_ I believe this problem is related to directional derivatives / gradient vectors but I don't know how to start with this. Thanks for your help.

So you know that the radius is increasing at the rate of 5 inches per second, so you can write $\frac{dr}{dt}=5$ and similarly we can write $\frac{dh}{dt}=-4$.

We know that the volume of a cone is $\frac{1}{3} \pi r^{2} h$, so to fine the rate of change of volume or $\frac{dV}{dt}$ we need to first express V as $\frac{1}{3} \pi r^{2} h$ and then use the product rule to write $\frac{dV}{dt}$ as something in terms of $r$, $h$, $\frac{dr}{dt}$ and $\frac{dh}{dt}$. Then you use the values of $r$ and $h$ at the values given (30 inches and 20 inches) and you will have found the rate of change of volume at the point where $r=30$ and $h=20$.

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