To determine how long the rope must be when the horse has moved through 60 cm at the extremity of the rope, and the angle traced out by the rope is 60 degrees, we need to understand the relationship between the arc length, radius of the circle (which in this case is the length of the rope), and the angle in radians.
1. Understand the relationship:
The formula relating the arc length [tex]\( s \)[/tex], the radius [tex]\( r \)[/tex] of the circle, and the angle [tex]\( \theta \)[/tex] in radians is given by:
[tex]\[
s = r \theta
\][/tex]
Here, [tex]\( s \)[/tex] is the arc length, [tex]\( \theta \)[/tex] is the angle in radians, and [tex]\( r \)[/tex] is the radius of the circle.
2. Given information:
- Arc length [tex]\( s \)[/tex] = 60 cm
- Angle [tex]\( \theta \)[/tex] = 60 degrees
3. Convert the angle from degrees to radians:
The angle in radians can be found using the conversion factor:
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\][/tex]
For our problem:
[tex]\[
\theta_{\text{radians}} = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
\][/tex]
Numerically this is approximately:
[tex]\[
\theta_{\text{radians}} \approx 1.0471975511965976
\][/tex]
4. Using the formula to find the radius [tex]\( r \)[/tex]:
Substitute the known values into the arc length formula [tex]\( s = r \theta \)[/tex]:
[tex]\[
60 = r \times 1.0471975511965976
\][/tex]
5. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{60}{1.0471975511965976}
\][/tex]
Numerically calculating this gives:
[tex]\[
r \approx 57.29577951308233 \text{ cm}
\][/tex]
Thus, the length of the rope must be approximately 57.29577951308233 cm in order for the horse to trace an angle of 60 degrees when it has moved through an arc length of 60 cm.
