Question

the radius of a circle is 5 miles. what is the length of a 45° arc?
give the exact answer in simplest form.
miles

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc 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.

Step2: Substitute values

Given \( r = 5 \) miles and \( \theta=45^{\circ} \), substitute these into the formula:
\( s=\frac{45^{\circ}}{360^{\circ}}\times2\pi\times5 \)

Step3: Simplify the fraction and multiply

Simplify \( \frac{45}{360}=\frac{1}{8} \), then:
\( s=\frac{1}{8}\times10\pi=\frac{5\pi}{4} \)

Answer:

\( \frac{5\pi}{4} \)