Question

the diameter of a circle is 10 kilometers. what is the angle measure of an arc 4π kilometers long?
l=4π km
d=10 km
give the exact answer in simplest form.
°

Explanation:

Step1: Calculate the circumference of the circle

The formula for the circumference of a circle is \(C=\pi d\). Given \(d = 10\) km, then \(C=\pi\times10=10\pi\) km.

Step2: Set up the proportion for the arc length and central angle

The proportion for the arc length \(l\), central angle \(\theta\) (in degrees), and circumference \(C\) is \(\frac{l}{C}=\frac{\theta}{360}\). We know \(l = 4\pi\) km and \(C=10\pi\) km. Substitute these values into the proportion: \(\frac{4\pi}{10\pi}=\frac{\theta}{360}\).

Step3: Solve for \(\theta\)

First, simplify \(\frac{4\pi}{10\pi}=\frac{4}{10}=\frac{2}{5}\). So, \(\frac{2}{5}=\frac{\theta}{360}\). Cross - multiply: \(5\theta=2\times360\). Then \(5\theta = 720\). Divide both sides by 5: \(\theta=\frac{720}{5}=144\).

Answer:

\(144\)