Chapter 12.4, Problem 19WE

a.

To determine

To find: The volume of the silo.

Answer to Problem 19WE

The volume of the silo is $\frac{1750\pi }{3}{\text{ m}}^3$ .

Explanation of Solution

Given information:

From the given figure height of the silo is 20 m and diameter of the silo is 10 m.

Calculation:

From the figure

  $\text{Volume of the silo}=\text{Volume of the cylindrical part }+\text{Volume of the hemispherical part}$

The Volume of cylinder is given by

  $V=\pi r^{2}h$ where $r$ is the radius of the cylinder and h is the height.

Also, volume of the hemisphere is given by

  $V=\frac{2}{3}\pi r^3$ where $r$ is the radius of the hemisphere.

Here, diameter of the silo is 10 m and height of the cylindrical part is 20 m . Therefore,

Radius

  $\begin{array}{l}r=\frac{10}{2}\\ r=5\text{ m}\end{array}$

Therefore,

  $\begin{array}{l}\text{Volume of the silo}=\pi r^{2}h+\frac{2}{3}\pi r^3\\ \text{Volume of the silo}=\pi {\left(5\right)}^2\left(20\right)+\frac{2}{3}\pi {\left(5\right)}^3\\ \text{Volume of the silo}=500\pi +\frac{250}{3}\pi \\ \text{Volume of the silo}=\frac{1500\pi +250\pi }{3}\\ \text{Volume of the silo}=\frac{1750\pi }{3}{\text{ m}}^3\end{array}$

Hence,

The volume of the silo is $\frac{1750\pi }{3}{\text{ m}}^3$ .

b.

To determine

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

Answer to Problem 19WE

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.