Chapter 13.7, Problem 24PSC

a

To determine

To find: The volume of the solid formed by rotating the triangle about OA.

Answer to Problem 24PSC

The volume of the solid formed by rotating the triangle about OA is $402.1\text{units}$ .

Explanation of Solution

Given information:

The co-ordinates of the triangle are $A\left(6,0\right),B\left(0,8\right),\text{and }O\left(0,0\right)$ .

If the triangle rotates about any side, then it form a cone.

The volume of the solid formed by rotating the triangle about OA is calculated as,

  $\begin{array}{c}V=\frac{1}{3}\pi r^{2}h\\ =\frac{1}{3}\pi {\left(8\right)}^2\left(6\right)\\ =402.1\text{units}\end{array}$

Thus, the volume of the solid formed by rotating the triangle about OA is $402.1\text{units}$ .

b

To determine

To find: The volume of the solid formed by rotating the triangle about OB.

Answer to Problem 24PSC

The volume of the solid formed by rotating the triangle about OB is $301.6\text{units}$ .

Explanation of Solution

Given information:

The co-ordinates of the triangle are $A\left(6,0\right),B\left(0,8\right),\text{and }O\left(0,0\right)$ .

If the triangle rotates about any side, then it forms a cone.

The volume of the solid formed by rotating the triangle about OB is calculated as,

  $\begin{array}{c}V=\frac{1}{3}\pi r^{2}h\\ =\frac{1}{3}\pi {\left(6\right)}^2\left(8\right)\\ =301.6\text{units}\end{array}$

Thus, the volume of the solid formed by rotating the triangle about OB is $301.6\text{units}$ .

c

To determine

To find: The volume and the total surface area of the solid formed by rotating the triangle about AB.

Answer to Problem 24PSC

The volume and the total surface area of the solid formed by rotating the triangle about AB.

are $241.3\text{units}$ and $211.1\text{units}$ .

Explanation of Solution

Given information:

The co-ordinates of the triangle are $A\left(6,0\right),B\left(0,8\right),\text{and }O\left(0,0\right)$ .

If the triangle rotates about any side, then it forms a cone.

The volume of the solid formed by rotating the triangle about AB is calculated as,

  $\begin{array}{c}V=\frac{1}{3}\pi r^2{h}_1+\frac{1}{3}\pi r^2{h}_2\\ =\frac{1}{3}\pi {\left(\frac{24}{5}\right)}^2\left(h_1+h_2\right)\\ =\frac{1}{3}\pi {\left(\frac{24}{5}\right)}^2\left(\sqrt{6^2+8^2}\right)\\ =241.3\text{units}\end{array}$

The total surface area of the solid formed by rotating the triangle about AB is calculated as,

  $\begin{array}{c}SA=\pi r{l}_1+\pi r{l}_2\\ =\pi \left(\frac{24}{5}\right)\left(6+8\right)\\ =211.1\text{units}\end{array}$

Thus, the volume and the total surface area of the solid formed by rotating the triangle about AB.

are $241.3\text{units}$ and $211.1\text{units}$ .