Question

1. a sector has an arc length of 15π cm and a central angle of 150°. find:

a) the radius of the circle
b) the area of the sector

1. a circular garden has a radius of 25 feet. a sprinkler in the center waters a sector with a central angle of 80°. what is the area of the lawn that gets watered?

round to nearest square foot

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \). Here, \( r = 25 \) feet and \( \theta=80^{\circ} \).

Step2: Substitute the values into the formula

Substitute \( r = 25 \) and \( \theta = 80^{\circ} \) into the formula:
\( A=\frac{80^{\circ}}{360^{\circ}}\times\pi\times(25)^{2} \)
Simplify \( \frac{80}{360}=\frac{2}{9} \) and \( (25)^{2}=625 \), so \( A=\frac{2}{9}\times\pi\times625 \)

Step3: Calculate the numerical value

First, calculate \( \frac{2\times625}{9}=\frac{1250}{9}\approx138.89 \)
Then multiply by \( \pi \): \( A\approx138.89\times\pi\approx138.89\times3.1416\approx436.19 \)

Step4: Round to the nearest square foot

Rounding \( 436.19 \) to the nearest whole number gives \( 436 \).

Answer:

The area of the lawn that gets watered is approximately \( 436 \) square feet.