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:

Question

A sector showing a quarter of a circle is drawn. The radius is marked as 80 feet. The endpoints of the arc of the sector are marked as Starting Point and Return Point. The sector is filled with cattle.

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? (5 points)

125.6 feet

502.4 feet

83.73 feet

62.8 feet

Answer 1

To calculate the distance the handler moves along the arc of a quarter circle with a radius of 80 feet, we can use the formula for the circumference of a circle, which is given by:

\[ C = 2\pi r \]

Since we are dealing with a quarter of a circle, we will take one-fourth of the total circumference:

1. Calculate the total circumference of the circle: \[ C = 2\pi \times 80 \text{ feet} = 160\pi \text{ feet} \]

2. Now calculate a quarter of that circumference: \[ \text{Arc Length} = \frac{1}{4} \times 160\pi \text{ feet} = 40\pi \text{ feet} \]

3. Using the approximation \(\pi \approx 3.14\): \[ 40\pi \approx 40 \times 3.14 = 125.6 \text{ feet} \]

Therefore, the distance the handler moves from the starting point to the return point is 125.6 feet.