Textbook Question
Find the length to three significant digits of each arc intercepted by a central angle in a circle of radius r. See Example 1. r = 12.3 cm , θ = 2π/3 radians
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1. Measuring Angles
Radians
1:45 minutesProblem 17
Textbook Question
Find the length to three significant digits of each arc intercepted by a central angle in a circle of radius r. See Example 1.
r = 4.82 m , θ = 60°
Verified step by step guidance1
Recall the formula for the length of an arc intercepted by a central angle in a circle: \(\text{Arc length} = r \times \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
Since the given angle \(\theta\) is in degrees, convert it to radians using the conversion formula: \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\).
Substitute the given values into the conversion formula: \(\theta_{\text{radians}} = 60 \times \frac{\pi}{180}\).
Calculate the arc length by multiplying the radius \(r = 4.82\) m by the angle in radians: \(\text{Arc length} = 4.82 \times \theta_{\text{radians}}\).
Express the final answer rounded to three significant digits as required.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a circle segment is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula relates the linear distance along the circle's edge to the angle subtended at the center.
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Conversion Between Degrees and Radians
Since the arc length formula requires the angle in radians, converting degrees to radians is essential. Use the conversion factor π radians = 180°, so θ (radians) = θ (degrees) × π / 180.
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Significant Figures in Measurement
When reporting the arc length, it is important to round the result to three significant digits to reflect the precision of the given data. This ensures the answer is both accurate and appropriately precise.
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