# Thread: Related rates with a cone, I just need verification.

1. ## Related rates with a cone, I just need verification.

"The volume of a cone of radius r and height h is given by v=1/3 pi r^2 h.If the radius and the height both increase at a constant rate of 1/2 centimeters per second, at what rate,in cubic centimeters per second, is the volume increasing when the height is 9 cm and the radius is 6 cm?"

Using the radius and height given, I first found the volume, which is 108 pi. Now from the problem, I know that dr/dt=1/2. So I would have to find the derivative of the volume, but I am still missing dh/dt. In order to solve for dh/dt, do I make dv/dt = 108 pi? I don't think so, but I'm trying really hard to solve these problems on my own.

"The volume of a cone of radius r and height h is given by v=1/3 pi r^2 h.If the radius and the height both increase at a constant rate of 1/2 centimeters per second, at what rate,in cubic centimeters per second, is the volume increasing when the height is 9 cm and the radius is 6 cm?"

Using the radius and height given, I first found the volume, which is 108 pi. Now from the problem, I know that dr/dt=1/2. So I would have to find the derivative of the volume, but I am still missing dh/dt. In order to solve for dh/dt, do I make dv/dt = 108 pi? I don't think so, but I'm trying really hard to solve these problems on my own.
$\displaystyle 108\pi$ is a constant. To get the derivative of $\displaystyle V$ with respect to time $\displaystyle t,$ you need to leave $\displaystyle V$ in terms of $\displaystyle t.$

$\displaystyle V=\frac13\pi r^2h$

$\displaystyle \Rightarrow\frac{dV}{dt}=\frac13\pi\left[r^2\frac{dh}{dt}+2rh\frac{dr}{dt}\right]$

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### the volume of a cone of radius r and height h is given by v=1/3pir^2h

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