To determine the length of the path described by the extremities of a pendulum, we can use the formula for the arc length of a circle segment. The formula for the arc length [tex]\( L \)[/tex] is given by:
[tex]\[ L = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle (in this case, the length of the pendulum) and [tex]\( \theta \)[/tex] is the central angle in radians. Here are the steps involved:
1. Identify the given values:
- Length of the pendulum [tex]\( r = 14 \)[/tex] cm.
- Angle [tex]\( \theta = 12^\circ \)[/tex].
2. Convert the angle from degrees to radians:
Recall that:
[tex]\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \][/tex]
Therefore, to convert degrees to radians, we use:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \][/tex]
Plugging in the given angle:
[tex]\[ \theta_{\text{radians}} = 12^\circ \times \frac{\pi}{180} = \frac{12\pi}{180} = \frac{\pi}{15} \text{ radians} \][/tex]
3. Apply the arc length formula:
Substitute [tex]\( r = 14 \)[/tex] cm and [tex]\( \theta = \frac{\pi}{15} \)[/tex] radians into the arc length formula:
[tex]\[ L = r \theta = 14 \times \frac{\pi}{15} \][/tex]
4. Simplify the expression:
[tex]\[ L = \frac{14 \pi}{15} \text{ cm} \][/tex]
Thus, the length of the path described by the extremities of the pendulum is:
[tex]\[ \boxed{\frac{14 \pi}{15} \text{ cm}} \][/tex]
Therefore, the correct answer is option (4): [tex]\(\frac{14 \pi}{15}\)[/tex].
