Q. 83 - 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 central angle $\theta$ is $0.24 radian.$

Step 1. Given information

$r=5 miles, A=3 square miles and \theta =?$

Step 2. Formula Used

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

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

Step 3. Calculation

We have,

$r=5 miles aand A=3 square miles$

Now,

The area $A$ of a sector is

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

$\Rightarrow 3=\frac{1}{2}\times 5^2\times \theta$

$\Rightarrow 3=\frac{25}{2}\theta$

$\Rightarrow \theta =0.24 radian$