Question

A spherical ball is immersed in water contained in a vertical cylinder. The rise in water level is measured in order to calculate the radius of the ball.Calculate the radius of the ball in the following cases:
(a): cylinder of radius 10cm, water level rises 4cm.
(b): cylinder of radius 100cm, water level rises 8cm

Answer 1

Here are the formulas for the cylinder and spheres respectively.
\[V_{cyl}=\pi*r^2*h\]
\[v_{sph}=\frac{4}{3}\pi*r^3\]
you know the the sphere, when immersed, increases the volume. By how much? Well the volume of the cylinder without the sphere is V=pi*r^2*h
The Volume of the cylinder afterwards is V=pi*r^2*(h+delta h), that means that the sphere must be the volume of the difference!
\[v_{sph}=v_{cyl+\delta*h}-v_{cyl}=\pi*r^2*(h+\delta*h)-\pi*r^2*h=\pi*r^2*\delta*h=\pi*(10cm)^2*(4cm)\]
\[v_{sph}=\frac{4}{3}\pi*r^3=\pi*(10cm)^2*(4cm)\]
Now, you can solve for the radius. (b) is a similar case.