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Charlie stands at point [tex]\( A \)[/tex] while holding the control line attached to a model airplane. The plane travels 120 feet counterclockwise from point [tex]\( B \)[/tex] to point [tex]\( C \)[/tex]. About how long is the control line?

The control line is about _______.



Answer :

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.