Question

Rates of Change question help. Will reward medal and fan. *Question attached below*

Answer 1

Looks as tho' the net amount of water entering the cone is (20-15) cubic centimeters per second.

The dimensions of the cone are, unfortunately, not known. We don't know the ratio of height to the diameter/radius of the open end of the cone.

The general formula for the volume of a right circular cone is \[V _{Cone}=\frac{ 1 }{3 }\pi*r^2h\]

Answer 2

Looks as tho' the net amount of water entering the cone is (20-15) cubic centimeters per second.

The dimensions of the cone are, unfortunately, not known. We don't know the ratio of height to the diameter/radius of the open end of the cone.

The general formula for the volume of a right circular cone is \[V _{Cone}=\frac{ 1 }{3 }\pi*R^2*H\] Here I use R for cone radius and H for cone height.

But what we're really interested in is not the volume of the cone itself, but rather the volume of the water inside the cone.

We want to know how fast the water depth is increasing when that depth is h=12 cm.

To make this problem easier up front, I'll assume that the height of the cone is always equal to the radius of the water surface (which is certainly not always the case). Then the equation for the volume of the water in the cone becomes
\[ V _{Cone}=\frac{ 1 }{3 }\pi*h^2*h= \frac{ 1 }{ 3 }\pi*h^3.\]

Finding the derivative of this latter equation with respect to time t introduces time rates of change:

Answer 3

\[\frac{ dV }{ dt }=\frac{ \pi }{ 3 }*3h^2\frac{ dh }{ dt? }\rightarrow \frac{ dV }{ dt }=\pi*h^2*\frac{ dh }{ dt }\]

Answer 4

All we have to do now is to substitute the knowns and calculate (dh/dt).

Questions?

Answer 5

Just give me a second to read everything and understand :)

Answer 6

Please note that this approach is dependent upon my assumption that h=r. You could, of course, use other assumptions, such as that h=2r, h=r/3, and so on.

Answer 7

Yes, I understand that

Answer 8

So, if you use an assumption like that, stating the assumption must occur along with your problem solution.

We know the rate of change of the volume of the water in the cone: It's (5 cm^3) / 1 sec.
We're interested in calculating how fast the water depth (h) is increasing when h=12 cm.

Substituting these values into the equation I've derived will give you your answer, the time rate of change of the water depth.

Answer 9

Thank you! I understand! :)

Answer 10

@mathmale What do I substitute dV/dt?

Answer 11

Looks as tho' the net amount of water entering the cone is (20-15) cubic centimeters per second, or 5 cc per sec. Try that.

Answer 12

I ended up with 5/144 pi. Is that correct?

Answer 13

I haven't tried doing this problem myself. You strike me as being very competent, so I'd bet you're right. Only thing I'd suggest is that you include units of measurement, which in this case are cm/sec. The depth of the water is increasing at that rate when the depth is 12 cm.

I will respond further, but will be unavailable for the next 30 minutes. Great working with you!

Answer 14

Thanks a lot! Great working with you too!