Chapter 12.4, Problem 21PSC

To determine

To calculate: The volume of the solid.

Answer to Problem 21PSC

The volume of solid is $90\pi$ .

Explanation of Solution

Given information:

Diameter of larger semicircle d = 12,

Diameter of two smaller semicircle d = 6,

Height of solid h =10.

Formula used:

Volume of prism $=B\times h$

Where

“B” is Base area of prism

“h” is height of prism

Area of a circle: $A=\pi r^2$

r is radius of circle

Calculation:

  

Area of larger semicircle:

d = 12

  $r=\frac{d}{2}=\frac{12}{2}=6$

  $\begin{array}{l}{A}_{LargerSemicircle}=\frac{1}{2}\pi r^2=\frac{1}{2}\pi {\left(6\right)}^2=18\pi \\ V_{LargerSemicircle}=A\times h\\ V_{LargerSemicircle}=18\pi \times 10\\ V_{LargerSemicircle}=180\pi \end{array}$

Area of smaller semicircle:

d = 6

  $r=\frac{d}{2}=\frac{6}{2}=3$

  $\begin{array}{l}{A}_{SmallerSemicircle}=\frac{1}{2}\pi r^2=\frac{1}{2}\pi {\left(3\right)}^2=4.5\pi \\ V_{SmallerSemicircle}=A\times h\\ V_{SmallerSemicircle}=4.5\pi \times 10\\ V_{SmallerSemicircle}=45\pi \end{array}$

There are two semicircles of radius 6.

Volume of prism $=V_{LargerSemicircle}-2\times V_{SmallerSemicircle}$

Volume of prism $=180\pi -\left(2\times 45\pi \right)$

Volume of prism $=180\pi -90\pi$

Volume of prism $=90\pi$