How is the Height of Water in a Container Changing Over Time?

[armolinasf]

Homework Statement

If there is a container with a radius in feet given by r(z)=10/sqrt(z) and it is being filled at a constant rate of 22pi ft^3/min. find a function that gives the height of the water as a function of time.

The Attempt at a Solution

since the radius is a function of the container is a function of z feet above the ground we integrate pir^2 => pi(10/sqrt(z))^2

100$$\pi$$$$\int^{H}_{0}$$1/z*dz

Applying the FTC we get v(H)=100pi*ln(H) making the v'(H)=100pi/H

We also know that the container is being filled at a rate of 22pi cubic feet per minute, this dv/dt

Since we know dv/dt and dv/dH we can create a related rate problem to solve for dH/dt:

dv/dt=dv/dH *dH/dt => 22pi=100pi/H*dH/dt => 11H/50=dH/dtThe way I interpret 11H/50=dH/dt is that if the height is H=10, for example, then the rate at which the height is increasing 110/50 ft/minute is this a correct interpretation?

My next question is regarding how I can go from dH/dt to H(t), that is if i know the time i can find how high the water level is.

My thinking is that I use: $$\int^{t}_{0}$$dH/dt*dH

so if Dh/dt is 11H/50 and I apply the FTC and evaluate on the interval o to T, where T is time in minutes I get H(T)=11t^2/100Does this make sense? or am I going wrong somewhere?

Thanks for the help

 

[Kreizhn]

It looks fine to me, though you switched notation from z to H. Also, don't forget your constant of integration. You need an additional piece of information: namely, the height at some initial time. I guess you've just assumed that at t=0 the height is zero.

 

[armolinasf]

There was one detail I omitted. Namely, that the container starts at z=1 foot, so i switched the notation from z to H to differentiate between the function r(z) and the actual container I'm trying to model which ranges z=H=1 to some height z or H. And yes at t=0 the height is zero.

I'm glad to hear that it makes sense: I really feel like I'm getting a good sense of calculus...