Question

1. an arc has length 8π inches and is intercepted by a central angle of 144°. what is the radius of the circle?

part 4: sector area

1. find the area of a sector with central angle 90° in a circle with radius 6 feet.

leave answer in terms of π

Explanation:

Question 6:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) is \( s=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. We know \( s = 8\pi \) inches and \( \theta=144^\circ \). Substitute these values into the formula: \( 8\pi=\frac{144^\circ}{360^\circ}\times2\pi r \).

Step2: Simplify the equation

First, simplify \( \frac{144^\circ}{360^\circ}=\frac{2}{5} \). So the equation becomes \( 8\pi=\frac{2}{5}\times2\pi r \), which simplifies to \( 8\pi=\frac{4\pi r}{5} \).

Step3: Solve for \( r \)

Multiply both sides of the equation by \( 5 \) to get \( 40\pi = 4\pi r \). Then divide both sides by \( 4\pi \): \( r=\frac{40\pi}{4\pi}=10 \).

Step1: Recall sector area formula

The formula for the area of a sector \( A \) is \( A=\frac{\theta}{360^\circ}\times\pi r^2 \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. Here, \( \theta = 90^\circ \) and \( r = 6 \) feet.

Step2: Substitute values into the formula

Substitute \( \theta = 90^\circ \) and \( r = 6 \) into the formula: \( A=\frac{90^\circ}{360^\circ}\times\pi\times(6)^2 \).

Step3: Simplify the expression

Simplify \( \frac{90^\circ}{360^\circ}=\frac{1}{4} \) and \( (6)^2 = 36 \). So \( A=\frac{1}{4}\times\pi\times36 = 9\pi \).

Answer:

The radius of the circle is \( \boldsymbol{10} \) inches.

Question 7: