Question

the radius of a circle is 3 kilometers. what is the area of a sector bounded by a 114° arc? give the exact answer in simplest form. square kilometers

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector is $A=\frac{\theta}{360}\times\pi r^{2}$, where $\theta$ is the central angle of the sector and $r$ is the radius of the circle.

Step2: Substitute the given values into the formula

Given that $\theta = 114^{\circ}$ and $r = 3$ km.
Substitute into the formula: $A=\frac{114}{360}\times\pi\times(3)^{2}$.
First, simplify $\frac{114}{360}=\frac{19}{60}$ (dividing numerator and denominator by 6).
Then, $(3)^{2}=9$.
So, $A=\frac{19}{60}\times\pi\times9$.

Step3: Calculate the value

$A=\frac{19\times9}{60}\pi=\frac{171}{60}\pi=\frac{57}{20}\pi$ (dividing numerator and denominator by 3).

Answer:

$\frac{57}{20}\pi$ square kilometers