Help with rate of change [Derivative]

Here is the problem:
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 centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?

I went ahead and differentiated it with respect to time t.

dV/dt = (2/3)dr/dt (h)(r)(pi) + (1/3)dh/dt(r^2)pi
dV/dt = 18pi + 6pi
dV/dt = 24pi

That's the correct way to do this problem I assume (since that's the answer my my textbook). Having said that, I tried to solve it in a different way.
Since the radius and the height change at a constant rate, I think they must have a fixed ratio.
So I put r/h = 6/9 <=> r = (2/3)h
Then I substitute that into the volume equation, and I got:
V = (4/27)(h^3)pi
dV/dt = (4/9)dh/dt(h^2)pi
dV/dt = (4/9)(1/2)(9^2)pi
dV/dt = 18pi

I don't know what went wrong there, everything seems perfectly logical. The radius and height increase at a constant speed so they would have the same ratio throughout the entire shape, wouldn't they?
Anyone have any idea what I did wrong?
Any idea will be appreciated.

Yo_2T said:

Here is the problem:
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 centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?

I went ahead and differentiated it with respect to time t.

dV/dt = (2/3)dr/dt (h)(r)(pi) + (1/3)dh/dt(r^2)pi
dV/dt = 18pi + 6pi
dV/dt = 24pi

That's the correct way to do this problem I assume (since that's the answer my my textbook). Having said that, I tried to solve it in a different way.
Since the radius and the height change at a constant rate, I think they must have a fixed ratio.

Click to expand...

You are mistaken. If h/r= constant, then h= constant r so dh/dt= constant times dr/dt. It does not follow that the constant is 1.

So I put r/h = 6/9 <=> r = (2/3)h

Click to expand...

But then dr/dt= (2/3)dh/dt. They cannot be the same.

Then I substitute that into the volume equation, and I got:
V = (4/27)(h^3)pi
dV/dt = (4/9)dh/dt(h^2)pi
dV/dt = (4/9)(1/2)(9^2)pi
dV/dt = 18pi

I don't know what went wrong there, everything seems perfectly logical. The radius and height increase at a constant speed so they would have the same ratio throughout the entire shape, wouldn't they?
Anyone have any idea what I did wrong?
Any idea will be appreciated.

Click to expand...

HallsofIvy said:

You are mistaken. If h/r= constant, then h= constant r so dh/dt= constant times dr/dt. It does not follow that the constant is 1.


But then dr/dt= (2/3)dh/dt. They cannot be the same.

Click to expand...

Thank you! I got it now