Chapter 12.3, Problem 17PPS

To determine

To find: the volume of the given figure

Answer to Problem 17PPS

The volume of the figure is $2628.2{\text{cm}}^3$ .

Explanation of Solution

Given:

  

Calculation:

The objective is to find the volume of the figure.

The volume of the figure is the sum of the volume of the rectangular box and the volume of the two half cylinder.

The volume V of the half cylinder is given by the formula

  $V=B\cdot h$

where $B=\frac{1}{2}\pi r^2$ is the area of the semicircle and h is the height the cylinder.

The volume V of a rectangular prism is given by

  $V=l\cdot w\cdot h$

wherel is the length, w is the width and w is the height of the rectangular prism.

Since radius is half the diameter of a circle, so radius of the circle is $\frac{8}{2}$ cm that is 4 cm.

The area B of the semicircle is given

  $\begin{array}{c}B=\frac{1}{2}\pi r^2\\ =\frac{1}{2}\pi {\left(4\right)}^2\\ =\frac{1}{2}\pi \cdot 16\\ =8\pi \end{array}$

Put $B=8\pi$ and h = 23 in the volume formula to find the volume.

Thus, the volume $V_1$ of the half cylinder is

  $\begin{array}{c}{V}_1=B\cdot h\\ =8\pi \cdot 23\\ =184\pi \\ \approx 184\left(3.14\right)\left[Since\pi \approx 3.14\right]\\ =578.128\end{array}$

The volume $V_2$ of the rectangular prism is

  $\begin{array}{c}{V}_2=l\cdot w\cdot h\\ =23\cdot 8\cdot 8\\ =1472\end{array}$

The required volume V is

  $\begin{array}{c}V=2V_1+V_2\\ \approx 2\left(578.128\right)+1472\\ =156.256+1472\\ \approx 2628.2\end{array}$

Therefore, the volume of the figure is $2628.2{\text{cm}}^3$ .

Conclusion:

Therefore, the volume of the figure is $2628.2{\text{cm}}^3$ .