Problem 12 - Precalculus Essentials

Question

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius, \(r\) 1 meter Arc Length, \(s\) 600 centimeters

Step-by-Step Solution

Answer

The central angle measures 6 radians.

Step 1: Conversion of Units

Firstly, we need to convert the length of the arc from centimeters to meters because in the formula, it is crucial that both the radius and arc length should be in the same units. Therefore, to convert from centimeters to meters, you should divide by 100 because there are 100 centimeters in a meter. Thus, \(s'=s/100=600/100=6\) meters.

Step 2: Calculate the Central Angle

After ensuring that the units are consistent, we can calculate the central angle by applying the formula: \(\theta=s'/r\), where \(\theta\) is the central angle, \(s'\) is the arc length and \(r\) is the radius of the circle. Substituting in the given values, we get: \(\theta=6/1=6\).

Step 3: Interpret the Result

The result we obtained is the measure of the central angle in radians. We must note that, the larger the radian measure, the larger the central angle. Hence, our result indicates that, for a circle of radius 1 meter and an arc length of 600 centimeters, the central angle measures 6 radians.

Key Concepts

Central Angle

The central angle in a circle is the angle subtended at the center by an arc. You can imagine it as the angle you get when you draw two lines from the center of the circle to the ends of the arc. It is a crucial concept when working with circles as it helps you determine how much of the circle the arc covers.
Central angles play a significant role in geometry and trigonometry since they directly relate to the arc's length and the circle's radius. In terms of radians, which is the most common measure in mathematics, the angle provides a more natural measure than degrees when calculating distances on the circle's circumference. A central angle's measure in radians is simply the arc's length divided by the radius of the circle. Therefore, \[\theta = \frac{s}{r}\]where \(s\) is the arc length and \(r\) is the radius. The radian measure allows us to compare arcs and angles straightforwardly.

Arc Length

Arc length refers to the distance along the curved line making up the arc. In simpler terms, imagine unrolling the arc into a straight line; that precise length is what we call the arc length. It depends upon the size of the angle subtended at the center and the radius of the circle.
To find the arc length when expressed in terms of radians, you use the formula:
\[s = \theta \times r\]Where \(s\) is the arc length, \(\theta\) is the central angle in radians, and \(r\) is the radius of the circle. This formula is simple to remember: just multiply the radian measure of the angle by the radius to get the arc length.

• If you know the arc length and radius, you can find the central angle by rearranging the formula to \(\theta = \frac{s}{r}\).
• This approach directly ties the geometry of the circle to its algebraic properties.

Unit Conversion

Unit conversion is an essential skill in mathematics, particularly when dealing with formulas involving units of measurement. For many problems, including those involving radian measures, having consistent units is critical to arriving at a correct solution.
In the original exercise, the arc length was given in centimeters, while the radius was in meters. Therefore, converting the arc length from centimeters to meters was necessary to maintain consistency because 100 centimeters equal 1 meter.

• To convert centimeters to meters, divide by 100: \(600\, \text{cm} = 6\, \text{m}\).
• Conversely, to convert meters to centimeters, multiply by 100: \(1\, \text{m} = 100\, \text{cm}\).

This conversion ensures that both the arc length and radius use the same unit, facilitating the use of mathematical formulas without introducing errors due to unit inconsistency. Remember, working in consistent units is a fundamental part of problem-solving in math and science.