Question

Three tennis balls are stored in a cylindrical container with a height of $8$ inches and a radius of $1.43$ inches.

The circumference of a tennis ball is $8$ inches.

Find the amount of space within the cylinder not taken up by the tennis balls?

Answer 1

Step-1: The volume of a tennis ball:

The circumference of the tennis ball is $8$ inches.

The tennis ball is the form of sphere whose circumference is given by formula $2\mathrm{\pi r}$, where $r$ is the radius.

Thus, if $r$ is the radius then according to given condition, $2\mathrm{\pi r}=8$ or $r=\frac{8}{2\mathrm{\pi }}$inches.

Now, the volume of the sphere of radius r is $\frac{4}{3}{\mathrm{\pi r}}^3$, hence, find the volume of the given tennis ball by substituting $r=\frac{8}{2\mathrm{\pi }}$inches in $\frac{4}{3}{\mathrm{\pi r}}^3$and simplify:

$$\begin{gathered} \text{Volume}=\frac{4}{3}\pi \times {\left(\frac{8}{2\mathrm{\pi }}\right)}^3 \\ =\frac{4}{3}\times \frac{64}{{\mathrm{\pi }}^2} \\ =8.65{\text{inches}}^3 \end{gathered}$$

Hence the required volume of the tennis ball is $8.65{\text{inches}}^3$.

The volume of three tennis balls is $\left(3\times 8.65\right){\text{inches}}^3=25.96{\text{inches}}^3$.

Step-2: Find the volume of the cylinder:

The volume of the cylinder with radius r units and height h units is given by ${\mathrm{\pi r}}^2\mathrm{h}$. Hence the volume of the given cylinder with radius $1.43$ inches , height $8$inches is:

$$\begin{gathered} =\mathrm{\pi }\times {\left(1.43\right)}^2\times 8 \\ =3.14\times 2.04\times 8 \\ =51.36{\text{inches}}^3 \end{gathered}$$

Hence the volume of the cylinder is $51.36{\text{inches}}^3$.

Step-3: Find the amount of space within the cylinder not taken up by the tennis balls.

The required volume can be obtained by subtracting the volume three tennis balls from the volume of the cylinder as follows:

$\text{Volume of cylinder}-\text{Volume of three tennis balls}=\left(51.36-25.96\right){\text{inches}}^3=25.40{\text{inches}}^3$

Hence, the amount of space within the cylinder not taken up by the tennis balls is $25.40{\text{inches}}^3$.