Definite integral distance from volume

[poopcaboose]

Homework Statement

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

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

 

[poopcaboose]

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

 

[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.

 

[Dick]

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

 

[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

 

[badphysicist]

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

 

[poopcaboose]

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