Question

the radius of a circle is 48 centimeters. what is the angle measure of an arc 46π centimeters long?
r=48 cm
l=46π cm
give the exact answer in simplest form.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( l \) of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( l=\frac{\theta}{360}\times2\pi r \).

Step2: Substitute known values

We know \( l = 46\pi \), \( r = 48 \). Substitute into the formula:
\( 46\pi=\frac{\theta}{360}\times2\pi\times48 \)

Step3: Simplify the equation

First, cancel \( \pi \) from both sides:
\( 46=\frac{\theta}{360}\times96 \)
Then, solve for \( \theta \):
\( \theta=\frac{46\times360}{96} \)
Simplify the fraction: \( \frac{46\times360}{96}=\frac{46\times15}{4}=\frac{690}{4}=\frac{345}{2} = 172.5\) (or as a fraction \( \frac{345}{2} \) or \( 172\frac{1}{2} \))
Wait, let's re - calculate the simplification:
\( \frac{46\times360}{96}=\frac{46\times15\times24}{4\times24}=\frac{46\times15}{4}=\frac{23\times15}{2}=\frac{345}{2}=172.5 \)
But let's do it step by step:
\( 46\pi=\frac{\theta}{360}\times96\pi \)
Divide both sides by \( \pi \): \( 46=\frac{\theta\times96}{360} \)
Cross - multiply: \( \theta\times96 = 46\times360 \)
\( \theta=\frac{46\times360}{96}=\frac{46\times15}{4}=\frac{690}{4}=\frac{345}{2}=172.5 \) (or \( 172\frac{1}{2} \) degrees)
Wait, let's check the calculation again:
\( 46\times360 = 16560 \)
\( 16560\div96=\frac{16560\div 24}{96\div24}=\frac{690}{4}=\frac{345}{2}=172.5 \)

Answer:

\( \frac{345}{2} \) (or \( 172.5 \) or \( 172\frac{1}{2} \)) degrees