Q. 81 - 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 $3.46$ feet.

Step 1. Given information

$\theta =\frac{1}{3}radians , A=2 square feet, r=?$

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,

$\theta =\frac{1}{3}radians , A=2 square feet$

Now, 

The area $A$ of a sector is

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

$\Rightarrow 2=\frac{1}{2}\times r^2\times \frac{1}{3}$

$$\begin{gathered} \Rightarrow 2=\frac{1}{6}{r}^2 \\ \Rightarrow r^2=12 \\ \Rightarrow r=\sqrt{12} \\ \Rightarrow r\approx 3.46 feet \end{gathered}$$