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Chapter 6: Problem 83

Find the length to three significant digits of each arc intercepted by a central angle \(\theta\) in a circle of radius \(r\). $$r=15.1 \text { in., } \theta=210^{\circ}$$

Short Answer

Expert verified
55.3 in

Step by step solution

01

Convert the angle to radians

First, convert the central angle \(\theta\) from degrees to radians using the formula \( \theta = \theta_{degrees} \times \frac{\pi}{180} \). For \(\theta = 210^{\circ}\), the calculation is: \(\theta = 210^{\circ} \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}\) radians.
02

Use the arc length formula

The formula for arc length \(s\) is \(s = r \theta\). Substitute the given \(r\) and the converted \(\theta\) into the formula: \( s = 15.1 \times \frac{7\pi}{6} \).
03

Calculate the arc length

Perform the multiplication to find the arc length: \( s = 15.1 \times \frac{7\pi}{6} = \frac{105.7\pi}{6} \).
04

Approximate the result

Approximate \( \pi \) as 3.14159 and calculate the numerical value: \( s = \frac{105.7 \times 3.14159}{6} = 55.3 \) in. Round the result to three significant digits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Angle
The central angle is the angle subtended by an arc at the center of a circle. In simpler terms, it is the angle formed at the circle's center by lines connecting the center to the ends of the arc. This angle is crucial for calculating arc length because it directly influences how much of the circle's circumference the arc covers. Understanding the central angle helps to visualize the portion of the circle you are working with. For instance, an angle of 210 degrees is more than half of the circle, which is 360 degrees. This means the arc covers a significant portion of the circle.
Radian Conversion
To calculate the arc length, it's often necessary to convert degrees into radians. Radians are another unit used to measure angles. They are especially useful in mathematics and science because they simplify many calculations. The formula to convert degrees to radians is: \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \).
Let's break it down with an example:
- If your central angle is 210 degrees, converting it to radians involves multiplying 210 by \( \frac{\pi}{180} \).
- So, \( 210^{\circ} \times \frac{\pi}{180} = \frac{7\pi}{6} \) radians.
This conversion is a foundational step in finding the arc length because the arc length formula uses radians, not degrees.
Arc Length Formula
Once the central angle is in radians, you can use the arc length formula to find the length of the arc. The arc length formula is: \( s = r \theta \),
where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. This formula is straightforward:
  • First, you need the circle's radius. In our case, it is 15.1 inches.
  • Next, take the central angle in radians \( \left( \frac{7\pi}{6} \right) \).
  • Multiply the radius by the angle in radians: \( 15.1 \times \frac{7\pi}{6} \).
This multiplication gives you the arc length in terms of \( \pi \). To find the exact numerical arc length, you then approximate \( \pi \) and perform the multiplication.
Significant Digits
When presenting a final answer, especially in scientific and mathematical contexts, it is important to round it to the correct number of significant digits. Significant digits reflect the precision of your measurement or calculation. Here's a quick guide to significant digits:
  • Non-zero digits are always significant.
  • Any zeros between significant digits are significant.
  • Leading zeros are not significant.
  • Trailing zeros are significant if they come after a decimal point.
In our exercise, the radius has three significant digits (15.1), so we should present our final arc length to three significant digits: 55.3 inches. Rounding to the correct number of significant digits ensures precision and reliability in your calculations, making your results scientifically accurate and trustworthy.

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