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 sector area formula

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 values into 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 value

First, calculate \( \frac{2\times625}{9}=\frac{1250}{9}\approx138.89 \). Then multiply by \( \pi \): \( 138.89\times\pi\approx138.89\times3.1416\approx436 \) (rounded to nearest square foot).

Answer:

The area of the watered region (sector) is approximately 436 square feet.