Chapter 13, Problem 3CT

(a)

To determine

To find: The length $s$ .

Answer to Problem 3CT

The length $s$ of intercepted arc is $=\frac{13\pi }{9}$

Explanation of Solution

Given information:

The central angle of a sector of a circle having diameter 8 in.measures ${65}^{°}$ .

Calculation:

The radius of a circle is half of diameter. So, the radius $r$ of the given sector is $\frac{8}{2}=4$ in.

We know that $1^{°}=\frac{\pi }{180}radians$ .

In order to convert the given degree measure to radians, multiply by $\frac{\pi }{180}$ .

  $\left(65\right).\frac{\pi }{180}radians=\frac{13\pi }{36}radians$

Now, the length of an arc is given by the formula $s=r\theta ,$ where $\theta$ is the radian measure, and $r$ is the radius of the circle.

Substitute $\frac{13\pi }{36}$ for $\theta$ , $4$ for $r$ in the formula, and simplify.

  $\begin{array}{c}s=\left(4\right)\left(\frac{13\pi }{36}\right)\\ =\frac{13\pi }{9}\end{array}$

Therefore, the length $s$ of intercepted arc is $=\frac{13\pi }{9}$

(b)

To determine

To find: Area $A$ of the sector.

Answer to Problem 3CT

The area of the sector $A$ is $\frac{26\pi }{9}$ .

Explanation of Solution

Given information:

The central angle of a sector of a circle having diameter 8 in.measures ${65}^{°}$ .

Calculation:

The area of a sector $A$ is given by the formula $A=\frac{1}{2}{r}^2\theta$ .

Substitute $\frac{13\pi }{36}$ for $\theta$ , $4$ for $r$ in the formula, and simplify.

  $\begin{array}{c}A=\frac{1}{2}{\left(4\right)}^2\left(\frac{13\pi }{36}\right)\\ =\frac{1}{2}(16)\left(\frac{13\pi }{36}\right)\\ =\frac{26\pi }{9}\end{array}$ .

Therefore the area of the sector $A$ is $\frac{26\pi }{9}$ .