Question

the diameter of a circle is 48 centimeters. what is the angle measure of an arc 19π centimeters long?
l=19π cm
d=48 cm
give the exact answer in simplest form.

Explanation:

Step1: Find the circumference of the circle

The formula for the circumference \( C \) of a circle is \( C=\pi d \), where \( d \) is the diameter. Given \( d = 48 \) cm, so \( C=\pi\times48 = 48\pi \) cm.

Step2: Use the arc length formula to find the angle

The arc length formula is \( l=\frac{\theta}{360^\circ}\times C \), where \( l \) is the arc length, \( \theta \) is the central angle in degrees, and \( C \) is the circumference. We know \( l = 19\pi \) cm and \( C = 48\pi \) cm. Substitute these values into the formula:
\( 19\pi=\frac{\theta}{360^\circ}\times48\pi \)
First, divide both sides by \( \pi \): \( 19=\frac{\theta}{360^\circ}\times48 \)
Then, solve for \( \theta \): \( \theta=\frac{19\times360^\circ}{48} \)
Simplify the fraction: \( \frac{19\times360}{48}=\frac{19\times15}{2}=\frac{285}{2}=142.5^\circ \) or \( \frac{285}{2}^\circ \) (which can also be written as \( 142\frac{1}{2}^\circ \))

Answer:

\( \frac{285}{2} \) (or \( 142.5 \)) degrees