Chapter 5.3, Problem 68E

a.

To determine

To prove:

The radius of the cone is,

  $r=\frac{1}{\sqrt{\pi }}\sqrt{S+\frac{\pi }{4}}-\frac{1}{2}$

Explanation of Solution

Given information:

The surface area of a right circular cone with a slant height of 1 unit is given as,

  $S=\pi r+\pi r^2$ , here $r$ is the radius of the cone

Proof:

Given,

  $S=\pi r+\pi r^2$

Now,

  $\begin{array}{l}S+\frac{\pi }{4}=\pi r+\pi r^2+\frac{\pi }{4}\left[\text{Add }\frac{\pi }{4}\text{ to both sides}\right]\\ S+\frac{\pi }{4}=\pi \left(r^2+r+\frac{1}{4}\right)\\ S+\frac{\pi }{4}=\pi {\left(r+\frac{1}{2}\right)}^2\\ \frac{S+\frac{\pi }{4}}{\pi }={\left(r+\frac{1}{2}\right)}^2\\ r+\frac{1}{2}=\pm \sqrt{\left(\frac{S+\frac{\pi }{4}}{\pi }\right)}\\ r+\frac{1}{2}=\pm \frac{1}{\sqrt{\pi }}\sqrt{S+\frac{\pi }{4}}\left[\text{Since }r>0\text{ take only positive value}\right]\\ r+\frac{1}{2}-\frac{1}{2}=\frac{1}{\sqrt{\pi }}\sqrt{S+\frac{\pi }{4}}-\frac{1}{2}\left[\text{Subtract }\frac{1}{2}\text{ from both sides}\right]\\ r=\frac{1}{\sqrt{\pi }}\sqrt{S+\frac{\pi }{4}}-\frac{1}{2}\left[\text{Hence proved}\right]\end{array}$

Therefore, the radius of the cone, $r=\frac{1}{\sqrt{\pi }}\sqrt{S+\frac{\pi }{4}}-\frac{1}{2}$ is proved.

b.

To determine

To calculate:

The radius of a right circular cone and to graph the equation obtained in part (a)

Answer to Problem 68E

The radius of the right circular cone is 0.5 and the graph is represented from the obtained data.

Explanation of Solution

Given information:

The slant height of the cone is 1 unit and the surface area of the cone is $\frac{3\pi }{4}$ square units.

Calculation:

The radius of the cone can be calculated as,

  $\begin{array}{l}r=\frac{1}{\sqrt{\pi }}\sqrt{S+\frac{\pi }{4}}-\frac{1}{2}\\ r=\frac{1}{\sqrt{\pi }}\sqrt{\frac{3\pi }{4}+\frac{\pi }{4}}-\frac{1}{2}\\ r=\frac{1}{\sqrt{\pi }}\sqrt{\frac{3\pi +\pi }{4}}-\frac{1}{2}\\ r=\frac{1}{\sqrt{\pi }}\sqrt{\frac{4\pi }{4}}-\frac{1}{2}\\ r=\frac{1}{\sqrt{\pi }}\times \sqrt{\pi }-\frac{1}{2}\\ r=1-\frac{1}{2}\\ r=\frac{2-1}{2}\\ r=\frac{1}{2}\\ r=0.5\end{array}$

Therefore, the radius of the right circular cone is 0.5

The graph of obtained equation can be represented as,