Question

the radius of a circle is 5 kilometers. what is the length of a 135° arc?
give the exact answer in simplest form.
kilometers

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc is \( L=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step2: Substitute values

Given \( \theta = 135^\circ \) and \( r = 5 \) km. Substitute these into the formula:
\( L=\frac{135^\circ}{360^\circ}\times2\pi\times5 \)

Step3: Simplify the fraction and multiply

Simplify \( \frac{135}{360}=\frac{3}{8} \). Then:
\( L=\frac{3}{8}\times10\pi=\frac{30\pi}{8}=\frac{15\pi}{4} \)

Answer:

\(\frac{15}{4}\pi\) (or \( \frac{15\pi}{4} \))