How Does Water Depth Change in a Conical Tank?

[Michele Nunes]

Homework Statement

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

Homework Equations

The Attempt at a Solution

Okay so I'm using the equation for the volume of a cone to relate the variables of volume and height: V = (1/3)πr²h, now I only want the variables for volume and height to be in the equation so I have to rewrite the radius r in terms of the height h which is where I'm unsure. I think that I should replace r with (5/12)h however someone else doing the problem replaced it with (12/5)h but I don't think that makes sense because you're saying the radius is 12/5 of the height which means it's bigger, however it's not, the radius is much smaller than the height which is why I think (5/12)h seems more logical. But once I rewrite r in terms of h, I've got the hang of it from there, it's just this one step that I'm unsure about.

 

[LCKurtz]

Homework Statement

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

Homework Equations

The Attempt at a Solution

Okay so I'm using the equation for the volume of a cone to relate the variables of volume and height: V = (1/3)πr²h, now I only want the variables for volume and height to be in the equation so I have to rewrite the radius r in terms of the height h which is where I'm unsure. I think that I should replace r with (5/12)h however someone else doing the problem replaced it with (12/5)h but I don't think that makes sense because you're saying the radius is 12/5 of the height which means it's bigger, however it's not, the radius is much smaller than the height which is why I think (5/12)h seems more logical. But once I rewrite r in terms of h, I've got the hang of it from there, it's just this one step that I'm unsure about.

$r = \frac 5 {12} h$ is correct.

 

[SteamKing]

Homework Statement

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

Homework Equations

The Attempt at a Solution

Okay so I'm using the equation for the volume of a cone to relate the variables of volume and height: V = (1/3)πr²h, now I only want the variables for volume and height to be in the equation so I have to rewrite the radius r in terms of the height h which is where I'm unsure. I think that I should replace r with (5/12)h however someone else doing the problem replaced it with (12/5)h but I don't think that makes sense because you're saying the radius is 12/5 of the height which means it's bigger, however it's not, the radius is much smaller than the height which is why I think (5/12)h seems more logical. But once I rewrite r in terms of h, I've got the hang of it from there, it's just this one step that I'm unsure about.

Making a simple sketch goes a long way to figuring out the answer to these kinds of questions.