Question

A ball of radius $13$ has a round hole of radius $6$ drilled through its center. find the volume of the resulting solid.

Answer 1

Determine the volume of the resulting solid:

Using the formulae for the volume of the sphere is $\frac{4}{3}{\mathrm{\pi r}}^3$ and volume of the cylinder is ${\mathrm{\pi r}}^2\mathrm{h}$.

The radius of a ball is $13$ and the radius of a round hole drilled through its center is $6$.

The volume of a sphere is as follows :

$$\begin{gathered} \mathrm{Volume}=\frac{4}{3}\times \frac{22}{7}\times {\left(13\right)}^3 \\ \mathrm{Volume}=\frac{88}{21}\times {\left(13\right)}^3 \\ \mathrm{Volume}=\frac{88}{21}\times 2197 \\ \mathrm{Volume}=\frac{193336}{21} \\ \mathrm{Volume}=9206.47\mathrm{cubic}\mathrm{units}. \end{gathered}$$

The drilled hole has the approximate shape of a solid cylinder with a radius is $6$ and its height is $13$.

The volume of the drilled hole is calculated as follows :

$$\begin{gathered} \mathrm{Volume}=\frac{22}{7}\times {\left(6\right)}^2\times 26 \\ \mathrm{Volume}=\frac{22}{7}\times 36\times 26 \\ \mathrm{Volume}=\frac{22}{7}\times 936 \\ \mathrm{Volume}=\frac{20592}{7} \\ \mathrm{Volume}=2941.71\mathrm{cubic}\mathrm{units} \end{gathered}$$

Thus, the resulting volume of the solid be

$$\begin{gathered} \Rightarrow V=9206.47-2941.71\mathrm{cubic}\mathrm{units} \\ \Rightarrow V=6264.76\mathrm{cubic}\mathrm{units}. \end{gathered}$$

Therefore, the resulting volume of the solid is $6264.76\mathrm{cubic}\mathrm{units}.$