Question

the diameter of a circle is 8 feet. what is the angle measure of an arc 3π feet long?
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 \(d = 8\) feet, then \(C=\pi\times8=8\pi\) feet.

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

The proportion for the arc length \(l\), circumference \(C\), central - angle \(\theta\) (in degrees), and \(360^{\circ}\) (full - circle angle) is \(\frac{l}{C}=\frac{\theta}{360}\).
Substitute \(l = 3\pi\) and \(C = 8\pi\) into the proportion: \(\frac{3\pi}{8\pi}=\frac{\theta}{360}\).
Since \(\pi\) cancels out (\(\frac{3\pi}{8\pi}=\frac{3}{8}\)), the equation becomes \(\frac{3}{8}=\frac{\theta}{360}\).

Step3: Solve for \(\theta\)

Cross - multiply: \(8\theta=3\times360\).
\(8\theta = 1080\).
Divide both sides by 8: \(\theta=\frac{1080}{8}=135\).

Answer:

\(135\)