Chapter 6.1, Problem 52E

a.

To determine

To find: the central angle in radians.

Answer to Problem 52E

  $3.4\text{ radians}$

Explanation of Solution

Given information:

  $\begin{array}{c}\text{Area of sector}\left(A\right)=15.3m^2\\ \text{Radius }\left(r\right)=3m\end{array}$

Calculation:

The central angle $\left(\theta \right)$ of the circle can be evaluated as follows:

  $\begin{array}{c}A=\frac{1}{2}{r}^2\theta \\ 15.3=\frac{1}{2}{\left(3\right)}^2\theta \left[\text{Put the value of }A\text{ and }r\right]\\ 15.3=\frac{9}{2}\theta \\ 9\theta =\left(15.3\right)\left(2\right)\\ \theta =\frac{\left(15.3\right)\left(2\right)}{9}\\ =3.4\text{ radians}\end{array}$

Thus the central angle of the circle is $3.4\text{ radians}$ .

b.

To determine

To find: the central angle in degrees.

Answer to Problem 52E

  $194.8°$

Explanation of Solution

Given information:

  $\text{Central angle}\left(\theta \right)=3.4\text{ radians}$

Calculation:

Convert radians into degree by multiplying the degree by $\frac{180°}{\pi }$ as shown below.

  $\begin{array}{c}\theta =3.4\left(\frac{180°}{\pi }\right)\\ =\frac{612°}{\pi }\\ =194.8°\end{array}$

Thus the radius of the circle in degree measure is $194.8°$ .

c.

To determine

To find: the arc length of the sector.

Answer to Problem 52E

  $10.2m$

Explanation of Solution

Given information:

  $\begin{array}{c}\text{Radius }\left(r\right)=3m\\ \text{Central angle}\left(\theta \right)=3.4\text{ radians}\end{array}$

Calculation:

The arc length of sector $\left(s\right)$ is evaluated as follows:

  $\begin{array}{c}s=r\theta \\ =\left(3\right)\left(3.4\right)\left[\text{Put the value of }r\text{ and }\theta \right]\\ =10.2m\end{array}$

Thus the arc length of the sector is $10.2m$ .