Answer :
Given the scenario where Charlie is holding the control line attached to a model airplane that travels 120 feet counterclockwise from point B to point C, we need to determine the length of the control line, which is essentially the radius of the circular path of the airplane.
Here’s the detailed step-by-step solution:
1. Understand the Relationship:
- The airplane travels along an arc of a circle.
- The distance it travels (120 feet) represents an arc length of that circle.
- We need to determine the length of the control line, which is the radius [tex]\( r \)[/tex] of the circle.
2. Use the Formula for the Circumference:
- The total circumference [tex]\( C \)[/tex] of a circle is given by [tex]\( C = 2\pi r \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.
- Since the plane travels along the circumference, we consider the distance traveled (120 feet) to be a part of this circumference.
3. Calculate the Radius:
- Given the distance traveled (120 feet) and knowing it forms part of the circle, we can rearrange the circumference formula to solve for the radius:
[tex]\[ r = \frac{\text{distance traveled}}{2\pi} \][/tex]
- Substituting the distance traveled into the formula:
[tex]\[ r = \frac{120}{2\pi} \][/tex]
4. Numerical Calculation:
- Plugging in the values and evaluating:
[tex]\[ r \approx \frac{120}{2 \times 3.14159} \approx 19.098593171027442 \text{ feet} \][/tex]
Hence, the length of the control line is approximately 19.1 feet. Thus, the correct answer to fill in the blank is:
The control line is about 19.1 feet long.
Here’s the detailed step-by-step solution:
1. Understand the Relationship:
- The airplane travels along an arc of a circle.
- The distance it travels (120 feet) represents an arc length of that circle.
- We need to determine the length of the control line, which is the radius [tex]\( r \)[/tex] of the circle.
2. Use the Formula for the Circumference:
- The total circumference [tex]\( C \)[/tex] of a circle is given by [tex]\( C = 2\pi r \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.
- Since the plane travels along the circumference, we consider the distance traveled (120 feet) to be a part of this circumference.
3. Calculate the Radius:
- Given the distance traveled (120 feet) and knowing it forms part of the circle, we can rearrange the circumference formula to solve for the radius:
[tex]\[ r = \frac{\text{distance traveled}}{2\pi} \][/tex]
- Substituting the distance traveled into the formula:
[tex]\[ r = \frac{120}{2\pi} \][/tex]
4. Numerical Calculation:
- Plugging in the values and evaluating:
[tex]\[ r \approx \frac{120}{2 \times 3.14159} \approx 19.098593171027442 \text{ feet} \][/tex]
Hence, the length of the control line is approximately 19.1 feet. Thus, the correct answer to fill in the blank is:
The control line is about 19.1 feet long.
