Question

A spherical tank in a petro-chemical plant has a radius of 5m.
Give an expression which, when integrated from y=-r to
y=d-r ,will give the volume of liquid in the tank where r is
the radius and d is the depth of liquid.
If liquid is entering the tank at a rate of 1500 litres per second and
the depth is 3m how long will it take to fill the tank? Give your
answer in seconds.

Answer 1

The volume of a thin slice of water is the area of the slice times its thickness, dy

At a depth of d, with d<r, we have
the radius of the surface of the water is x = √(r^2-y^2)
v = ∫[-r,d-r] π(r^2-y^2) dy

Now you can answer the last part by subtracting the volume from the whole sphere's volume of 4/3 pi * 5^3 and dividing by 1500L/s = 1.5 m^3/s