Help finding the radius of a cylinder to obtain increase of volume

[mathhelp10001]

So I've been trying this problem for hours and I cant figure it out. I feel like I'm on the right track though.

I've been given: a cylinder with height of "x", a radius of "r(x)", a rate of filling the cylinder of f(t).


The question asks me to find the radius in ordder to get a rate of increase of h(t).


We were using a method from the differential equations chapter of seperable and linear equations paired with the method for finding volume from calculus 1.


Note: there are no numbers in this problem, only variables.


Any help would be greatly appreciated.

 

[Deleted member 4993]

So I've been trying this problem for hours and I cant figure it out. I feel like I'm on the right track though.

I've been given: a cylinder with height of "x", a radius of "r(x)", a rate of filling the cylinder of f(t). for a cylinder 'r' should be constant


The question asks me to find the radius in ordder to get a rate of increase of h(t).


We were using a method from the differential equations chapter of seperable and linear equations paired with the method for finding volume from calculus 1.


Note: there are no numbers in this problem, only variables.


Any help would be greatly appreciated.

V = π * r² * h

Now what ....

What else is given?

 

[mathhelp10001]

V = π * r² * h

Now what ....

What else is given?

I've included everything thats given. The height and the rate of change of the increasing volume. Now I need to find the radius of the cylinder at any given time(represented by h(t)).

This question is supposed to use separable equations I think.

 

[Deleted member 4993]

I've included everything thats given. The height and the rate of change of the increasing volume. Now I need to find the radius of the cylinder at any given time(represented by h(t)).

This question is supposed to use separable equations I think.

If it is a cylinder - by definition the "radius" is constant along the axis. The rate of change is zero (0).

 

[HallsofIvy]

Subhotosh Khan, I think you are misunderstanding the question. "r(x)" does NOT mean the radius is changing along the axis of the cylinder because x is NOT a variable measuring distance along the axis. This refers to a variable cylinders having length x and radius, r(x), some function of the total length. For example, if r(x)= x/2, then the cylinder always has radius half its length. The radius is changing as the length changes, not "along" the length.

However, I still can't make out what the "rate of filling" has to do with the "rate of increase of h(t)". I suspect, although it isn't said, that the height, h, of the cylinder, is the height of the water in it at each moment. In that case, the given "rate of filling" is the rate of increase of volume of the cylinder. Given a cylinder of height h(t) (your x) and a radius r(h), then the Volume is, as Subhotosh Khan said, \(\displaystyle V= \pi r^2h\) so its rate of change is \(\displaystyle dV/dt= f(t)= \pi(2r dr/t h+ r^2 dh/dt)= \pi(2r (dr/dh)h+ r^2)(dh/dt)\).