Question

the diameter of a circle is 10 meters. what is the angle measure of an arc 2π meters long?
d=10 m
l=2π m
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 \( d = 10 \) meters, so \( C=\pi\times10 = 10\pi \) meters.

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 \( l = 2\pi \) and \( C = 10\pi \), substituting these values in, we get \( \frac{2\pi}{10\pi}=\frac{\theta}{360^\circ} \).

Step3: Solve for \( \theta \)

Simplify \( \frac{2\pi}{10\pi}=\frac{1}{5} \). Then \( \frac{1}{5}=\frac{\theta}{360^\circ} \), cross - multiply to get \( \theta=\frac{360^\circ}{5}=72^\circ \).

Answer:

\( 72 \)