Question

name: ak lam
date:
period:
problem set #2
show all work.

1. the area of a sector of a circle with a radius measuring 15 cm is 75π cm². what is the measure of the central angle that forms the sector?

(1) 180°
(2) 120°
(3) 72°
(4) 144°

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $A$ is the area of the sector, $\theta$ is the central - angle measure in degrees, and $r$ is the radius of the circle.
We are given that $A = 75\pi\mathrm{cm}^{2}$ and $r = 15\mathrm{cm}$.

Step2: Substitute the given values into the formula

Substitute $A = 75\pi$ and $r = 15$ into $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$:
$75\pi=\frac{\theta}{360^{\circ}}\times\pi\times(15)^{2}$

Step3: Simplify the equation

First, simplify the right - hand side: $(15)^{2}=225$, so the equation becomes $75\pi=\frac{\theta}{360^{\circ}}\times225\pi$.
Divide both sides of the equation by $\pi$ (since $\pi
eq0$), we get $75=\frac{\theta}{360^{\circ}}\times225$.

Step4: Solve for $\theta$

Multiply both sides of the equation by $360^{\circ}$ to get $75\times360^{\circ}=\theta\times225$.
$75\times360 = 27000$, so $27000 = 225\theta$.
Then divide both sides by 225: $\theta=\frac{27000}{225}=120^{\circ}$

Answer:

$120^{\circ}$