Expand your horizons with the diverse and informative answers found on IDNLearn.com. Join our community to receive prompt and reliable responses to your questions from experienced professionals.
Sagot :
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].
[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].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.