Question

the radius of a circle is 32 meters. what is the angle measure of an arc 20π meters long?
r=32 m
l=20π m
give the exact answer in simplest form.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( l \) 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 known values

We know \( l = 20\pi \) meters and \( r = 32 \) meters. Substitute these into the formula:
\( 20\pi=\frac{\theta}{360^\circ}\times2\pi\times32 \)

Step3: Solve for \( \theta \)

First, simplify the right - hand side: \( \frac{\theta}{360^\circ}\times64\pi \)
Then, divide both sides by \( \pi \): \( 20=\frac{\theta}{360^\circ}\times64 \)
Next, multiply both sides by \( 360^\circ \): \( 20\times360^\circ = 64\theta \)
\( 7200^\circ=64\theta \)
Now, divide both sides by 64: \( \theta=\frac{7200^\circ}{64}=\frac{225^\circ}{2} = 112.5^\circ \)

Answer:

\( \frac{225}{2} \) (or \( 112.5 \))