Question

the diameter of a circle is 50 inches. what is the angle measure of an arc 21π inches long? 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 \( d = 50 \) inches, so \( C=\pi\times50 = 50\pi \) inches.

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

The ratio of the arc length \( s \) to the circumference \( C \) is equal to the ratio of the central angle \( \theta \) (in degrees) to \( 360^\circ \). That is \( \frac{s}{C}=\frac{\theta}{360^\circ} \). We know \( s = 21\pi \) and \( C = 50\pi \), so substitute these values in: \( \frac{21\pi}{50\pi}=\frac{\theta}{360^\circ} \). The \( \pi \) cancels out, giving \( \frac{21}{50}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

Cross - multiply: \( 50\theta=21\times360^\circ \). Then \( \theta=\frac{21\times360^\circ}{50} \). Simplify \( \frac{21\times360}{50}=\frac{21\times36}{5}=\frac{756}{5}=151.2^\circ \)? Wait, no, wait. Wait, \( 21\times360 = 7560 \), then \( \theta=\frac{7560}{50}=\frac{756}{5}=151.2 \)? Wait, no, that's wrong. Wait, no, wait, the formula for arc length in degrees is \( s=\frac{\theta}{360^\circ}\times2\pi r \), but since \( d = 50 \), \( r=\frac{d}{2}=25 \). Alternatively, using \( s = r\theta \) (when \( \theta \) is in radians), but we can also use the proportion. Wait, let's redo step 2. Wait, the arc length formula is \( s=\frac{\theta}{360^\circ}\times C \), where \( C = 2\pi r=\pi d \). So \( s=\frac{\theta}{360^\circ}\times\pi d \). We have \( s = 21\pi \), \( d = 50 \). So \( 21\pi=\frac{\theta}{360^\circ}\times\pi\times50 \). Divide both sides by \( \pi \): \( 21=\frac{\theta\times50}{360} \). Then \( \theta=\frac{21\times360}{50}=\frac{21\times36}{5}=\frac{756}{5}=151.2 \)? Wait, no, that can't be. Wait, no, 21360 = 7560, 7560/50 = 151.2? But that's a decimal. Wait, maybe I made a mistake. Wait, no, let's check again. Wait, the diameter is 50, so radius is 25. The circumference is \( 2\pi r=50\pi \), correct. The arc length is \( 21\pi \). So the fraction of the circle is \( \frac{21\pi}{50\pi}=\frac{21}{50} \). Then the angle is \( \frac{21}{50}\times360=\frac{21\times360}{50}=\frac{21\times36}{5}=\frac{756}{5}=151.2 \) degrees? But that's a decimal. Wait, but the problem says "exact answer in simplest form". Wait, 756 divided by 5 is 151.2, but maybe I messed up the formula. Wait, no, the formula for arc length is \( s = r\theta \) where \( \theta \) is in radians. Let's try that. \( r = 25 \), \( s = 21\pi \). So \( 21\pi=25\theta \), so \( \theta=\frac{21\pi}{25} \) radians. To convert to degrees, multiply by \( \frac{180^\circ}{\pi} \). So \( \theta=\frac{21\pi}{25}\times\frac{180^\circ}{\pi}=\frac{21\times180}{25}=\frac{21\times36}{5}=\frac{756}{5}=151.2^\circ \). Wait, but 756/5 is 151.2, which is \( \frac{756}{5} \) degrees or 151.2 degrees. But let's check the calculation again. 21180 = 3780, 3780/25 = 151.2. Yes. Wait, but maybe the problem expects a fractional degree. Wait, 756/5 is equal to 151 and 1/5 degrees, which is 151.2 degrees. Wait, but let's check the proportion again. The arc length is 21π, circumference is 50π. So the ratio is 21/50. So the angle is (21/50)360 = (21360)/50 = (21*36)/5 = 756/5 = 151.2 degrees. So that's the angle.

Answer:

\(\frac{756}{5}\) (or \(151.2\))