A 9-ft pendulum swings through an angle of 40° 36'. What is the length of the arc that the tip of the pendulum travels? Round intermediate calculations to 3 decimal places. Round your final answer to the nearest hundredth of a foot. The length of the arc is approximately ft. X Ś
A 9-ft pendulum swings through an angle of 40° 36'. What is the length of the arc that the tip of the pendulum travels? Round intermediate calculations to 3 decimal places. Round your final answer to the nearest hundredth of a foot. The length of the arc is approximately ft. X Ś
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter19: Achievement Review—section One
Section: Chapter Questions
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![**Pendulum Arc Length Calculation**
*Problem Statement:*
A 9-ft pendulum swings through an angle of \( 40^\circ 36' \). What is the length of the arc that the tip of the pendulum travels? Round intermediate calculations to three decimal places. Round your final answer to the nearest hundredth of a foot.
*Step-by-Step Solution:*
1. **Convert the Angle to Decimal Degrees:**
- \( 1^\circ = 60' \)
- Given angle: \( 40^\circ 36' \)
- Convert 36 minutes to degrees:
\[
36' = 36/60^\circ = 0.6^\circ
\]
- Sum of degrees:
\[
40^\circ + 0.6^\circ = 40.6^\circ
\]
2. **Convert Degrees to Radians:**
- Formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
\[
40.6^\circ \times \frac{\pi}{180} \approx 0.708 \text{ radians}
\]
3. **Calculate Arc Length:**
- Arc length formula: \( s = r\theta \)
- \( r \) is the radius of the circle (pendulum length): \( r = 9 \) ft
- \( \theta \) is the angle in radians: \( \theta \approx 0.708 \)
\[
s = 9 \times 0.708 \approx 6.374 \text{ ft}
\]
4. **Round the Final Answer:**
- Round \( s \) to the nearest hundredth:
\[
6.37 \text{ ft}
\]
*Final Answer:*
The length of the arc is approximately \( 6.37 \) ft.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6cc73a6-72c8-4e2b-a22b-5b06a5805dfc%2Ffee120a3-1ef1-4714-8c7b-6724d4c634be%2F0303uj9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Pendulum Arc Length Calculation**
*Problem Statement:*
A 9-ft pendulum swings through an angle of \( 40^\circ 36' \). What is the length of the arc that the tip of the pendulum travels? Round intermediate calculations to three decimal places. Round your final answer to the nearest hundredth of a foot.
*Step-by-Step Solution:*
1. **Convert the Angle to Decimal Degrees:**
- \( 1^\circ = 60' \)
- Given angle: \( 40^\circ 36' \)
- Convert 36 minutes to degrees:
\[
36' = 36/60^\circ = 0.6^\circ
\]
- Sum of degrees:
\[
40^\circ + 0.6^\circ = 40.6^\circ
\]
2. **Convert Degrees to Radians:**
- Formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
\[
40.6^\circ \times \frac{\pi}{180} \approx 0.708 \text{ radians}
\]
3. **Calculate Arc Length:**
- Arc length formula: \( s = r\theta \)
- \( r \) is the radius of the circle (pendulum length): \( r = 9 \) ft
- \( \theta \) is the angle in radians: \( \theta \approx 0.708 \)
\[
s = 9 \times 0.708 \approx 6.374 \text{ ft}
\]
4. **Round the Final Answer:**
- Round \( s \) to the nearest hundredth:
\[
6.37 \text{ ft}
\]
*Final Answer:*
The length of the arc is approximately \( 6.37 \) ft.
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