Question

question 6 (multiple choice worth 5 points) (07.03 mc) 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. 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? \\(\bigcirc\\) 125.6 feet \\(\bigcirc\\) 502.4 feet \\(\bigcirc\\) 80.73 feet \\(\bigcirc\\) 62.8 feet question 7 (multiple choice worth 5 points) (07.03 mc)

Explanation:

Step1: Identify the arc type

The figure shows a quarter - circle arc (since the triangle formed is a right - triangle, and the arc is a quarter of a circle). The formula for the length of an arc of a circle is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central angle in radians. For a quarter - circle, the central angle $\theta=\frac{\pi}{2}$ radians (or $90^{\circ}$, and in radians, $90^{\circ}=\frac{\pi}{2}$) and $r = 80$ feet. Also, the formula for the length of a quarter - circle arc can be derived from the circumference formula $C = 2\pi r$. The length of a quarter - circle arc $s=\frac{1}{4}\times2\pi r=\frac{\pi r}{2}$.

Step2: Substitute the radius into the formula

Substitute $r = 80$ feet into the formula $s=\frac{\pi r}{2}$. We get $s=\frac{\pi\times80}{2}=40\pi$.

Step3: Calculate the numerical value

We know that $\pi\approx3.14$, so $40\pi\approx40\times3.14 = 125.6$ feet. Wait, but let's check again. Wait, maybe the angle is a right angle (90 degrees) which is $\frac{1}{4}$ of a full circle (360 degrees). The circumference of a circle is $C = 2\pi r$. For a quarter - circle, the arc length $s=\frac{1}{4}C=\frac{1}{4}\times2\pi r=\frac{\pi r}{2}$. If $r = 80$, then $s=\frac{\pi\times80}{2}=40\pi\approx125.6$? Wait, but the options have 125.6 feet as one of the options. Wait, but maybe I made a mistake. Wait, no, let's re - check. Wait, the triangle is a right - triangle with legs of length 80 (since the radius is 80). Wait, maybe the arc is a quarter - circle, so the central angle is 90 degrees. So arc length $=\frac{\theta}{360}\times2\pi r$, where $\theta = 90^{\circ}$. So $\frac{90}{360}\times2\pi\times80=\frac{1}{4}\times160\pi = 40\pi\approx125.6$ feet.

Answer:

125.6 feet (the option with 125.6 feet)