Question

a circle with circumference 8 has an arc with a 288° central angle. what is the length of the arc?

Explanation:

Step1: Set up proportion

The ratio of the central - angle of the arc to the full - circle angle is equal to the ratio of the arc length to the circumference. The full - circle angle is $360^{\circ}$. Let the arc length be $l$ and the circumference $C = 8$. So, $\frac{l}{C}=\frac{\theta}{360^{\circ}}$, where $\theta = 288^{\circ}$.

Step2: Substitute values

Substitute $C = 8$ and $\theta = 288^{\circ}$ into the proportion $\frac{l}{8}=\frac{288^{\circ}}{360^{\circ}}$.

Step3: Solve for $l$

Cross - multiply: $l=\frac{288^{\circ}}{360^{\circ}}\times8$. Simplify $\frac{288}{360}=\frac{4}{5}$, then $l=\frac{4}{5}\times8=\frac{32}{5} = 6.4$.

Answer:

$6.4$