1. ## Spherical tanks

Consider a sphere shaped tank. Suppose it has radius 2. It is filled 2/3 of the way with slime. What is the depth of the slime at its deepest point (i.e. what is the height of the slime in the tank when the tank is 2/3 of the way full). Use calculus methods.

Next, compute the work to pump the slime out the top of tank that is 2/3 full, using the density of slime as 1000 kg/m^3

I know you do 4/3 pi r^3 for the volume of the tank, and so you can find the volume of 2/3 the tank, but how about the height?

Consider a sphere shaped tank. Suppose it has radius 2. It is filled 2/3 of the way with slime. What is the depth of the slime at its deepest point (i.e. what is the height of the slime in the tank when the tank is 2/3 of the way full). Use calculus methods.

Next, compute the work to pump the slime out the top of tank that is 2/3 full, using the density of slime as 1000 kg/m^3

I know you do 4/3 pi r^3 for the volume of the tank, and so you can find the volume of 2/3 the tank, but how about the height?
I assume you mean 2/3 full by volume.

$\pi \int_{-2}^h 4-y^2 \, dy = \frac{2}{3} \cdot \frac{4}{3} \pi \cdot 2^3$

solve for $h$

$W = 9800 \pi \int_{-2}^h (4 - y^2)(2 - y) \, dy$