Question

select the correct answer. circle o has radius of 24 units. arc xy located on the circle has a central angle of 75°. what is the area of the associated sector, in square units? a. 10π b. 120π c. 5π d. 51π

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \).

Step2: Identify the given values

We are given that \( r = 24 \) units and \( \theta=75^{\circ} \).

Step3: Substitute the values into the formula

First, calculate \( r^{2} \): \( r^{2}=24^{2} = 576 \).
Then, substitute into the sector area formula: \( A=\frac{75^{\circ}}{360^{\circ}}\times\pi\times576 \).

Step4: Simplify the expression

Simplify \( \frac{75}{360}\times576 \).
First, simplify \( \frac{75}{360}=\frac{5}{24} \).
Then, \( \frac{5}{24}\times576 = 5\times24=120 \).
So, \( A = 120\pi \).

Answer:

B. \( 120\pi \)