3650.14 – Tennis Balls

In a can of tennis balls that is exactly three balls high, which is greater, the volume of the balls or the volume of the air around them?

By "the volume of the balls" we mean the volume of the air inside the balls plus the volume of the material of which the balls are made.

Solution

The figure shows a cross section of the can.

The volume of one ball is $\dfrac{4\pi{r^3}}{3}$ , so all three have volume $4\pi{r^3}$ .

The volume of the cylinder is $\pi{r^2}\cdot6r = 6\pi{r^3}$ .

Thus the volume of air around the balls is $2\pi{r^3}$ , half the volume of the three balls.