To solve the given problem, we need to determine the length of the control line, which is the radius of the circular path that the model airplane follows.
1. Distance Traveled: The airplane travels 120 feet along a circular path.
2. Circumference of the Circle: The distance of 120 feet is a part of the circumference of the circle. To find the full circumference (C) of the entire path, we double the distance traveled (since a circular path has symmetric properties). Thus, the full circumference is:
[tex]\[
C = 2 \times 120 \text{ feet} = 240 \text{ feet}
\][/tex]
3. Formula for Circumference: The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\(r\)[/tex] is the radius (the length of the control line).
4. Solving for the Radius: Rearrange the formula to solve for the radius [tex]\(r\)[/tex]:
[tex]\[
r = \frac{C}{2 \pi}
\][/tex]
6. Substitute the Known Circumference: Substitute [tex]\(C = 240\)[/tex] feet into the formula:
[tex]\[
r = \frac{240 \text{ feet}}{2 \pi}
\][/tex]
7. Calculate the Radius: Perform the division:
[tex]\[
r \approx 38.197186342054884 \text{ feet}
\][/tex]
Therefore, the length of the control line is approximately 38.2 feet.
