Question

a circle with circumference 12 has an arc with a 48° central angle. what is the length of the arc?

Explanation:

Step1: Recall arc length formula

The length of an arc is a fraction of the circumference. The fraction is the ratio of the central angle to \(360^\circ\). So the formula is \( \text{Arc Length} = \frac{\theta}{360^\circ} \times \text{Circumference} \), where \( \theta \) is the central angle.

Step2: Substitute values

Here, \( \theta = 48^\circ \) and circumference \( = 12 \). Substitute into the formula: \( \text{Arc Length} = \frac{48^\circ}{360^\circ} \times 12 \)

Step3: Simplify the fraction

Simplify \( \frac{48}{360} \) by dividing numerator and denominator by 24: \( \frac{48\div24}{360\div24}=\frac{2}{15} \)

Step4: Calculate the arc length

Multiply \( \frac{2}{15} \times 12 \). \( \frac{2\times12}{15}=\frac{24}{15} = \frac{8}{5}=1.6 \)

Answer:

\( 1.6 \) (or \( \frac{8}{5} \))