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:
A sector showing a quarter of a circle is drawn. The radius is marked as 75 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 75 feet?
Answers
GPT-4o mini
To find the distance the handler moves from the starting point to the return point along the arc of a quarter of a circle with a radius of 75 feet, we can use the formula for the circumference of a circle and then calculate the length of the arc.
1. **Calculate the Circumference of the Full Circle:**
The formula for the circumference \( C \) of a circle is:
\[
C = 2\pi r
\]
where \( r \) is the radius.
Here, the radius \( r = 75 \) feet.
So,
\[
C = 2\pi(75) = 150\pi \text{ feet}
\]
2. **Calculate the Length of the Quarter Circle Arc:**
Since the handler is moving along a quarter of the circle, we need to take one-fourth of the full circumference:
\[
\text{Arc Length} = \frac{1}{4} \times C = \frac{1}{4} \times 150\pi = \frac{150\pi}{4} = 37.5\pi \text{ feet}
\]
3. **Approximate the Arc Length:**
To get a numerical approximation, we can use \( \pi \approx 3.14 \):
\[
\text{Arc Length} \approx 37.5 \times 3.14 \approx 117.75 \text{ feet}
\]
Thus, the handler will move approximately **117.75 feet** from the starting point to the return point along the arc of the circle.
1. **Calculate the Circumference of the Full Circle:**
The formula for the circumference \( C \) of a circle is:
\[
C = 2\pi r
\]
where \( r \) is the radius.
Here, the radius \( r = 75 \) feet.
So,
\[
C = 2\pi(75) = 150\pi \text{ feet}
\]
2. **Calculate the Length of the Quarter Circle Arc:**
Since the handler is moving along a quarter of the circle, we need to take one-fourth of the full circumference:
\[
\text{Arc Length} = \frac{1}{4} \times C = \frac{1}{4} \times 150\pi = \frac{150\pi}{4} = 37.5\pi \text{ feet}
\]
3. **Approximate the Arc Length:**
To get a numerical approximation, we can use \( \pi \approx 3.14 \):
\[
\text{Arc Length} \approx 37.5 \times 3.14 \approx 117.75 \text{ feet}
\]
Thus, the handler will move approximately **117.75 feet** from the starting point to the return point along the arc of the circle.