Question

the area of a sector of a circle with a central angle of 4 radians is 13ft². find the radius of the circle. do not round any intermediate computations. round your answer to the nearest tenth.

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $A$ is the area of the sector, $r$ is the radius of the circle, and $\theta$ is the central - angle in radians.
We are given that $A = 13$ square feet and $\theta=4$ radians.
Substitute these values into the formula: $13=\frac{1}{2}r^{2}\times4$.

Step2: Simplify the equation

First, simplify the right - hand side of the equation. $\frac{1}{2}\times4r^{2}=2r^{2}$. So our equation becomes $13 = 2r^{2}$.

Step3: Solve for $r^{2}$

Divide both sides of the equation by 2: $r^{2}=\frac{13}{2}=6.5$.

Step4: Solve for $r$

Take the square root of both sides. Since $r$ represents the radius of a circle (a non - negative quantity), $r=\sqrt{6.5}$.
Using a calculator, $r\approx2.5$ feet.

Answer:

$2.5$