Chapter 12.4, Problem 18WE

a.

To determine

To find: The volume to the correct nearest cubic centimetre.

Answer to Problem 18WE

The volume of the sphere to the nearest cubic centimetre is 113.04 cm².

Explanation of Solution

Given information:

The sphere is inscribed in the cube of edge 6 cm long.

Calculation:

The Volume sphere is given by

  $V=\frac{4}{3}\pi r^3$ where $r$ is the radius of the sphere.

The side view when the sphere is inscribed in cube is drawn below:

  

From, the above figure the radius of the inscribed sphere will be

  $\begin{array}{l}r=\frac{6}{2}\\ r=3\text{ cm}\end{array}$

Now, find volume of the sphere:

  $\begin{array}{l}V=\frac{4}{3}\pi {\left(3\right)}^3\\ V=\frac{4}{3}\times 3.14\times 27\\ V=113.04{\text{ cm}}^3\end{array}$

Hence,

The volume of the sphere to the nearest cubic centimetre is 113.04 cm³.

b.

To determine

To find: The volume of the region inside the cube but outside the sphere.

Answer to Problem 18WE

Volume of the region inside the cube but outside the sphere is 102.96 cm³.

Explanation of Solution

Given information:

The sphere is inscribed in the cube of edge 6 cm long.

Calculation:

Volume of the region inside the cube but outside the sphere is

  $\text{Volume of the cube}-\text{Volum of the sphere}$

The volume of the cube is given by

  $V_c=a^3$ , where a is the edge of the cube.

Volume of the cube is

  $\begin{array}{l}{V}_c={\left(6\right)}^3\\ V_c=216{\text{ cm}}^3\end{array}$

Also, from part (a) volume of the inscribed sphere is $V=113.04{\text{ cm}}^3$ .

Therefore, volume of the region inside the cube but outside the sphere is

  $\begin{array}{l}216-113.04\\ 102.96{\text{ cm}}^3\end{array}$

Hence,

Volume of the region inside the cube but outside the sphere is 102.96 cm³.