Question

the diameter of a circle is 10 miles. what is the angle measure of an arc π miles long?
l=π mi
d=10 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 \( d = 10 \) miles, so \( C=\pi\times10 = 10\pi \) 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 \( l = \pi \) miles and \( C = 10\pi \) miles. Substituting these values, we get \( \frac{\pi}{10\pi}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

Simplify \( \frac{\pi}{10\pi}=\frac{1}{10} \). Then \( \frac{1}{10}=\frac{\theta}{360^\circ} \). Cross - multiplying gives \( 10\theta=360^\circ \), so \( \theta = \frac{360^\circ}{10}=36^\circ \).

Answer:

\( 36 \)