Question

the figure below shows the ideal pattern of movement of a herd of cattle, with the arrows showing the movement of the handler as he moves the herd. the arc the handler makes from the starting point to the return point should be a quarter of a circle: direction of desired movement 80 ft starting point return point based on this theory, what distance will the handler move from the starting point to the return point if he creates an arc of a circle with radius 80 feet?

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc of a circle is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians.

Step2: Convert the central - angle to radians

Since the arc is a quarter of a circle, the central angle $\theta=\frac{1}{4}\times2\pi=\frac{\pi}{2}$ radians, and the radius $r = 80$ feet.

Step3: Calculate the arc - length

Substitute $r = 80$ and $\theta=\frac{\pi}{2}$ into the arc - length formula $s=r\theta$. So $s=80\times\frac{\pi}{2}=40\pi$ feet.

Answer:

$40\pi$ feet