Chapter 12.5, Problem 3ST2

To determine

Tofind:The volume of the time capsule consisting of a cylinder and twoheispheres.

Answer to Problem 3ST2

The volume of the time capsule shown is $\frac{22000\pi }{3}{\text{cm}}^3$ .

Explanation of Solution

Given information:

The students of a school decide to bury a time capsule consisting of a cylinder capped by two hemispheres.

  

Calculation:

The volume of the time capsule is the sum of the volume of cylinder and two hemispheres.

The diameter of the cylinder and sphere is $20\text{cm}$ . The radius is always the half of the diameter.So, the radius of cylinder and hemisphere is:

  $\frac{20\text{cm}}{2}=10\text{cm}$

The formula for the volume of hemisphere is $\text{V}=\frac{2}{3}\pi r^3$ .Substitute $10$ for $r$ in the formula of volume.

  $\begin{array}{c}\text{V}=\frac{2}{3}\pi {\left(10\text{cm}\right)}^3\\ =\frac{2}{3}\pi \times 1000{\text{cm}}^3\\ =\frac{2000}{3}\pi {\text{cm}}^3\end{array}$

The volume of one hemisphere is $\frac{2000}{3}\pi {\text{cm}}^3$ . So, the volume of both the hemispheres is:

  $2\times \frac{2000}{3}\pi {\text{cm}}^3=\frac{4000}{3}\pi {\text{cm}}^3$

Now, the height of the cylinder is $60\text{cm}$ and the radius of cylinder is $10\text{cm}$ .

The formula for the volume of the cylinder is $\text{V}=\pi r^{2}h$ .

Substitute $60$ for $h$ and $10$ for $r$ in the formula for volume of cylinder.

  $\begin{array}{c}\text{V}=\pi {\left(10\text{cm}\right)}^2\left(60\text{cm}\right)\\ =\pi \times 100{\text{cm}}^2\times 60\text{cm}\\ =\text{6000}\pi {\text{cm}}^3\end{array}$

The total volume of the time capsule can be calculated as:

  $\begin{array}{c}\text{V}=\frac{4000\pi }{3}{\text{cm}}^3+6000\pi {\text{cm}}^3\\ =\frac{4000\pi +18000\pi }{3}{\text{cm}}^3\\ =\frac{22000\pi }{3}{\text{cm}}^3\end{array}$

Therefore, the volume of the time capsule shown is $\frac{22000\pi }{3}{\text{cm}}^3$ .