Chapter 11, Problem 4CA

(a)

To determine

To write: an equation that relates $\theta$ and $x$ .

Answer to Problem 4CA

The required equation is $x\theta =900$

Explanation of Solution

Given:

  

Formula Used:

The arc length of sector is $2\pi r\cdot \frac{\theta }{{360}^{\circ }}$

Calculation:

Use the formula for arc length of sector

Are length = $2\pi r\cdot \frac{\theta }{{360}^{\circ }}$

  $\begin{array}{l}5\pi =2\pi \left(x\right)\cdot \frac{\theta }{{360}^{\circ }}\\ \frac{5}{2}\cdot 360=x\theta \\ \boxed{x\theta =900}\end{array}$

Conclusion:

The required equation is $x\theta =900$

(b)

To determine

To write: an equation that relates $\theta andx+10$ .

Answer to Problem 4CA

The required equation is $\left(x+10\right)\theta =2520$

Explanation of Solution

Calculation:

Use the formula for arc length of sector

Are length = $2\pi r\cdot \frac{\theta }{{360}^{\circ }}$

  $\begin{array}{l}14\pi =2\pi \left(x+10\right)\cdot \frac{\theta }{{360}^{\circ }}\\ \frac{14}{2}\cdot 360=\left(x+10\right)\theta \\ \boxed{\left(x+10\right)\theta =2520}\end{array}$

Conclusion:

The required equation is $\left(x+10\right)\theta =2520$

(c)

To determine

To solve: the system of equation for $xand\theta$ .

Answer to Problem 4CA

The required result is ${162}^{\circ }$

Explanation of Solution

Calculation:

To solve the system of equations from part (a) and (b), that is the system

  $\begin{array}{l}x\theta =900\\ \left(x+10\right)\theta =2520\end{array}$

First solve $x\theta =900$ for $\theta$

  $\begin{array}{l}x\theta =900\\ \theta =\frac{900}{x}\end{array}$

Now, substitute this value in

  $\begin{array}{l}\left(x+10\right)\theta =2520\\ \left(x+10\right)\frac{900}{x}=2520\\ 900x+9000=2520x\\ =9000=2520x-900x\\ 9000=1620x\\ x=\frac{9000}{1620}\\ x=5.5\overline{5}\end{array}$

Then,

  $\begin{array}{l}\theta =\frac{900}{x}\\ =\frac{900}{5.5\overline{5}}\\ ={162}^{\circ }\end{array}$

Conclusion:

The required result is ${162}^{\circ }$

(d)

To determine

To find: the amount of fabric that used to make the lampshade.

Answer to Problem 4CA

The amount of fabric needed is 298.45 square inches of fabric.

Explanation of Solution

Calculation:

The angle of the sector in radians is

  $\begin{array}{l}\theta ={162}^{\circ }\\ ={162}^{\circ }\left(\frac{\pi }{{180}^{\circ }}\right)\\ =\frac{9\pi }{10}\end{array}$

Area of the larger sector

  $\begin{array}{l}A=\frac{1}{2}{r}^2\theta \\ =\frac{1}{2}{\left(x+10\right)}^2\left(\frac{9\pi }{10}\right)\\ =\frac{1}{2}{\left(\frac{9000}{1620}+10\right)}^2\left(\frac{9\pi }{10}\right)\\ =\frac{1}{2}{\left(\frac{140}{9}\right)}^2\left(\frac{9\pi }{10}\right)\\ =\frac{980\pi }{9}\end{array}$

Area of smaller sector

  $\begin{array}{l}A=\frac{1}{2}{r}^2\theta \\ =\frac{1}{2}{\left(\frac{50}{9}\right)}^2\left(\frac{9\pi }{10}\right)\\ =\frac{125}{9}\pi \end{array}$

Thus, area of the lampshade is

  $\begin{array}{l}\begin{array}{l}Areaoflargersector -Areaofsmallersector=\frac{980}{9}\pi -\frac{125}{9}\pi \hfill \\ =95\pi \hfill \end{array}\\ =298.45\end{array}$

Thus, the amount of fabric needed is 298.45 square inches of fabric.

Conclusion:

Thus, the amount of fabric needed is 298.45 square inches of fabric.