Question

the diameter of a circle is 8 miles. what is the angle measure of an arc 3π miles long?
l=3π mi

d=8 mi

give the exact answer in simplest form.

Explanation:

Step1: Find the circumference of the circle

The formula for the circumference \( C \) of a circle is \( C = \pi d \), where \( d \) is the diameter. Given \( d = 8 \) miles, so \( C=\pi\times8 = 8\pi \) miles.

Step2: Set up the proportion for arc length and angle

The ratio of the arc length \( l \) to the circumference \( C \) is equal to the ratio of the central angle \( \theta \) (in degrees) to \( 360^\circ \). So \( \frac{l}{C}=\frac{\theta}{360^\circ} \). We know \( l = 3\pi \) and \( C = 8\pi \), substituting these values in, we get \( \frac{3\pi}{8\pi}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

First, simplify \( \frac{3\pi}{8\pi}=\frac{3}{8} \). Then we have \( \frac{3}{8}=\frac{\theta}{360^\circ} \). Cross - multiply: \( \theta=\frac{3\times360^\circ}{8} \). Calculate \( 3\times360 = 1080 \), then \( \frac{1080}{8}=135^\circ \).

Answer:

\( 135 \)