Q. 82 - Precalculus Enhanced with Graphing Utilities

Question

In the problem, $A$denotes the area of the sector of a circle of radius $r$ formed by the central angle $\theta$. Find the missing quantity. Round answers to three decimal places.

Step-by-Step Solution

Answer

The required radius of the circle is $6.93 centimeters$.

Step 1. Given information

$\theta =\frac{1}{4}radian, A=6 square centimeters, r=?$

Step 2. Formula Used

The area $A$ of the sector of a circle of radius $r$ formed by the central angle of  $\theta$ radians is

$A=\frac{1}{2}{r}^2\theta$.

Step 3. Calculation

We have,

$\theta =\frac{1}{4} radian and A=6 square centimeters$

Now, 

The Area $A$ of the sector is

$A=\frac{1}{2}{r}^2\theta$

$\Rightarrow 6=\frac{1}{2}\times r^2\times \frac{1}{4}$

$\Rightarrow 6=\frac{1}{8}{r}^2$

$\Rightarrow r^2=48$

$$\begin{gathered} \Rightarrow r=\sqrt{48} \\ \Rightarrow r\approx 6.93 centimeters \end{gathered}$$