# Definite integral distance from volume

1. Jul 26, 2008

### poopcaboose

1. The problem statement, all variables and given/known dataIf you have a water tower that is spherical with a radius 20m, how far from the bottom will the water level be if it is filled to 1/4 of it's capacity.

2. Relevant equations$$\int_-20^20pi(400-y^2)dy=33510.32164m^3=3,351,032.164Liters$$

3. The attempt at a solution I can find 1/4 capacity=837,758.041 liters but don't know where to go from here to find the distance from the bottom

2. Jul 26, 2008

### poopcaboose

I meant for it to be a definite integral from 20 to -20 not 20 squared

3. Jul 26, 2008

### Dick

Instead of integrating from -20 to 20 to get the full volume, integrate from -20 to h. You'll get a cubic expression in h. Equate that to 1/4 of the full volume and try to solve for h.

4. Jul 26, 2008

### Dick

BTW it looks like you have to use numerical techniques to get an answer. The cubic doesn't factor or anything for me.

5. Jul 26, 2008

### poopcaboose

Thanks for the help, I can use newtons method to approximate f(y) at zero and solve it from there thanks I would have never thought to set it to y

6. Jul 27, 2008

By the way, two of the roots are complex numbers from what I got.. just a warning.

7. Jul 27, 2008

### poopcaboose

I figured it out its 13.054073m from the bottom you either solve for zero with newtons method or graph it