To determine the length of the control line, which is the radius of the circular path, we need to use the relationship between the arc length and the radius of a circle. Here are the steps:
1. Understand the arc length formula:
The arc length [tex]\( s \)[/tex] of a circle is given by the formula:
[tex]\[
s = r \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius (length of the control line) and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Given data:
- The arc length [tex]\( s \)[/tex] is 120 feet.
3. Finding the central angle:
Because the arc length [tex]\( s \)[/tex] is a part of a circle and we don't know the exact central angle [tex]\( \theta \)[/tex], we will use the relationship that [tex]\(\theta\)[/tex] in radians can be related to the complete circle [tex]\(2\pi\)[/tex] and the fraction of the circle traveled.
Let's assume that the plane travels any arbitrary [tex]\( \theta \)[/tex] radians.
4. Determining the radius [tex]\( r \)[/tex]:
If we know [tex]\( s = 120 \)[/tex] feet, we can rearrange the arc length formula to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{s}{\theta}
\][/tex]
5. Assuming a 120 feet arc defines a fraction of the circle, let us consider [tex]\( \theta\)[/tex]:
We have [tex]\( s = 120 \)[/tex]
and
[tex]\( s \)[/tex] also equals [tex]\( 120 = r \theta \)[/tex]
6. Finding specific value for convenience, [tex]\(\theta = \frac{s}{120}\)[/tex]
Let's solve for [tex]\( r \)[/tex]:
1. Rewriting the equation:
The arc length [tex]\( s = 120 \)[/tex] feet is directly proportional to the radius [tex]\( r \)[/tex] through [tex]\( 2 \pi \)[/tex]:
[tex]\(\theta = \frac{s}{120}\)[/tex]
Solving equation without exact central angle assume to solve for radius:
[tex]\[
r = \frac{120}{2\pi}
\][/tex]
\[r = \frac{120}{6.28} \approx 19.1
Thus, the length of the control line is approximately 19 feet.
So, the final answer in the drop-down is:
The control line is about "19" feet long.
