To find the length of the control line Charlie is holding, we need to understand some key pieces of information from the problem:
1. Charlie stands at point A, and the airplane travels 120 feet from point B to point C along an arc that is part of a full circle.
2. This arc length (120 feet) represents a fraction of the entire circle's circumference.
Given that the 120 feet represents a certain fraction of the circle's total circumference, let's assume it's one-fourth (1/4) of the circle since it travels counterclockwise in this way. This means that the full circumference of the circle is:
[tex]\[ \text{Circumference} = 120 \, \text{feet} \times 4 \][/tex]
So the total circumference of the circle is 480 feet.
The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = 2 \pi \times \text{radius} \][/tex]
We know the circumference is 480 feet. By solving for the radius (which represents the length of the control line), we have:
[tex]\[ \text{radius} = \frac{\text{Circumference}}{2 \pi} \][/tex]
By substituting the circumference value:
[tex]\[ \text{radius} = \frac{480}{2 \pi} \][/tex]
We can now find the radius value:
[tex]\[ \text{radius} \approx \frac{480}{6.2832} \approx 76.39 \, \text{feet} \][/tex]
Therefore, the length of the control line Charlie is holding is about 76.39 feet.
From the given options in the drop-down menu:
The control line is about \( \boxed{76.39437268410977} \) feet long.
Since we need a close answer, and the provided one is not exactly present in the options, we can approximate it to the closest option:
The control line is about [tex]\( 86 \)[/tex] feet long.
