Question

a circle has a radius of 4. an arc in this circle has a central angle of 288°. what is the length of the arc? either enter an exact answer in terms of π or use 3.14 for π and enter your answer as a decimal.

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. First, convert the angle from degrees to radians. The conversion formula is $\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}$. Given $\theta_{deg}=288^{\circ}$, then $\theta_{rad}=\frac{\pi}{180}\times288=\frac{8\pi}{5}$.

Step2: Calculate the arc - length

We know that $r = 4$ and $\theta=\frac{8\pi}{5}$. Substitute these values into the arc - length formula $s=r\theta$. So $s = 4\times\frac{8\pi}{5}=\frac{32\pi}{5}$.
If we use $\pi = 3.14$, then $s=\frac{32\times3.14}{5}=\frac{100.48}{5}=20.096$.

Answer:

$\frac{32\pi}{5}$ or $20.096$