Question

1. if a circles sector has an area of 25π square cm and the central angle is π/2 radians, what is the radius?

a. 5 cm
b. 12 cm
c. 20 cm
d. 10 cm

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area \( A \) of a sector with central angle \( \theta \) (in radians) and radius \( r \) is \( A=\frac{1}{2}\theta r^{2} \).

Step2: Substitute the given values into the formula

We are given that \( A = 25\pi \) and \( \theta=\frac{\pi}{2} \). Substituting these into the formula:

$$ 25\pi=\frac{1}{2}\times\frac{\pi}{2}\times r^{2} $$

Step3: Simplify the equation

First, simplify the right - hand side: \( \frac{1}{2}\times\frac{\pi}{2}\times r^{2}=\frac{\pi r^{2}}{4} \). So our equation becomes:

$$ 25\pi=\frac{\pi r^{2}}{4} $$

We can divide both sides of the equation by \( \pi \) (since \( \pi
eq0 \)):

$$ 25 = \frac{r^{2}}{4} $$

Step4: Solve for \( r \)

Multiply both sides of the equation by 4:

$$ r^{2}=25\times4 = 100 $$

Then take the square root of both sides. Since \( r>0 \) (radius is a non - negative quantity), we have \( r = \sqrt{100}=10 \).

Answer:

d. 10 cm