Question

the diameter of a circle is 48 meters. what is the angle measure of an arc 5π meters long? give the exact answer in simplest form.

Explanation:

Step1: Find the circumference of the circle

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

Step2: Set up the proportion for arc length and angle

The ratio of the arc length to the circumference is equal to the ratio of the central angle (let it be \( \theta \)) to \( 360^\circ \). So \( \frac{\text{Arc Length}}{C}=\frac{\theta}{360^\circ} \). We know the arc length is \( 5\pi \) meters and \( C = 48\pi \) meters. Substituting these values, we get \( \frac{5\pi}{48\pi}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

First, simplify \( \frac{5\pi}{48\pi}=\frac{5}{48} \). Then, from \( \frac{5}{48}=\frac{\theta}{360^\circ} \), we can solve for \( \theta \) by cross - multiplying: \( \theta=\frac{5\times360^\circ}{48} \). Simplify \( \frac{5\times360}{48}=\frac{1800}{48}=\frac{75}{2}=37.5^\circ \) or in fraction form \( \frac{75}{2}^\circ \) (or \( 37\frac{1}{2}^\circ \)). But we can also keep it as a fraction. Let's do the calculation again: \( \theta=\frac{5\times360}{48}=\frac{5\times15}{2}=\frac{75}{2} \) (since \( 360\div24 = 15 \) and \( 48\div24 = 2 \)).

Answer:

\(\frac{75}{2}\) (or \(37.5\))