Question

the diameter of a circle is 50 kilometers. what is the angle measure of an arc 3π kilometers long?
d=50 km
l=3π km
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 \( d = 50 \) km, so \( C=\pi\times50 = 50\pi \) km.

Step2: Set up the proportion for arc length and angle

The ratio of the arc length \( l \) to the circumference \( C \) is equal to the ratio of the central angle \( \theta \) (in degrees) to \( 360^\circ \). So \( \frac{l}{C}=\frac{\theta}{360^\circ} \). We know \( l = 3\pi \) and \( C = 50\pi \), substituting these values: \( \frac{3\pi}{50\pi}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

Simplify \( \frac{3\pi}{50\pi}=\frac{3}{50} \). Then \( \frac{3}{50}=\frac{\theta}{360^\circ} \). Cross - multiply: \( 50\theta=3\times360^\circ \). So \( 50\theta = 1080^\circ \). Then \( \theta=\frac{1080^\circ}{50}=\frac{108^\circ}{5} = 21.6^\circ \)? Wait, no, wait. Wait, \( 3\times360 = 1080 \), \( 1080\div50=\frac{108}{5}=21.6 \)? Wait, no, wait, let's check again. Wait, \( \frac{3\pi}{50\pi}=\frac{3}{50} \), so \( \theta=\frac{3}{50}\times360^\circ=\frac{1080}{50}=\frac{108}{5}=21.6^\circ \)? Wait, no, that can't be. Wait, no, wait, the formula for arc length in degrees is \( l=\frac{\theta}{360^\circ}\times2\pi r \), and since \( d = 50 \), \( r = 25 \). So \( l=\frac{\theta}{360^\circ}\times2\pi r \), \( 3\pi=\frac{\theta}{360^\circ}\times2\pi\times25 \). Let's solve this way. Cancel \( \pi \) from both sides: \( 3=\frac{\theta}{360^\circ}\times50 \). Then \( \frac{\theta}{360^\circ}=\frac{3}{50} \), so \( \theta=\frac{3\times360^\circ}{50}=\frac{1080^\circ}{50}=\frac{108^\circ}{5}=21.6^\circ \)? Wait, but let's do it with radians first and then convert. The arc length formula is \( l = r\theta \) (in radians). \( r = 25 \), \( l = 3\pi \), so \( 3\pi=25\theta \) (in radians), \( \theta=\frac{3\pi}{25} \) radians. To convert to degrees: \( \theta=\frac{3\pi}{25}\times\frac{180^\circ}{\pi}=\frac{3\times180^\circ}{25}=\frac{540^\circ}{25}=\frac{108^\circ}{5}=21.6^\circ \). Wait, but \( \frac{108}{5}=21.6 \), but let's check the proportion again. \( \frac{3\pi}{50\pi}=\frac{3}{50} \), \( \frac{3}{50}\times360=\frac{1080}{50}=21.6 \). Yes, that's correct. Wait, but maybe I made a mistake in the first approach. Wait, no, the two methods (using \( C=\pi d \) and using \( l = r\theta \) with conversion) should agree. So the angle measure is \( \frac{108}{5}^\circ \) or \( 21.6^\circ \)? Wait, no, wait, \( 3\times360 = 1080 \), \( 1080\div50 = 21.6 \). Yes.

Wait, but let's re - express \( \frac{108}{5}^\circ=21.6^\circ \), but let's check the calculation again. \( \frac{3}{50}\times360=\frac{3\times360}{50}=\frac{1080}{50}=21.6 \). Yes. So the angle measure is \( \frac{108}{5} \) degrees or \( 21.6 \) degrees. But let's see, \( 3\pi \) arc length, circumference is \( 50\pi \). So the fraction of the circle is \( \frac{3\pi}{50\pi}=\frac{3}{50} \), so the angle is \( \frac{3}{50}\times360=\frac{1080}{50}=\frac{108}{5}=21.6 \) degrees.

Answer:

\( \frac{108}{5}^\circ \) (or \( 21.6^\circ \))