Problem 72 - Precalculus Essentials

Question

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\boldsymbol{\theta}\). Express arc length in terms of \(\pi .\) Then round your answer to two decimal places. Radius, \(r\) 16 inches Central Angle, \(\theta=60^{\circ}\)

Step-by-Step Solution

Answer

The length of the arc intercepted by a central angle of \(60^{\circ}\) on a circle of radius \(16\) inches is 16.755 rounded to two decimal places.

Step 1: Conversion of the Central Angle to Radians

The central angle is given in degrees. So, we must convert it to radians first. We know that \(360^{\circ}\) = \(2\pi\) radians, so the conversion can be done using this relation. Hence, \(\theta_{radians} = \theta_{degrees} \times \frac{2\pi radians}{360^{\circ}}\). Put the value of \(\theta = 60^{\circ}\) into this relation to calculate the value of \(\theta\) in radian.

Step 2: Expression for the Arc Length

The formula for the length of an arc is given by \(arc length = r \times \theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle measured in radians. It is important to keep in mind that \(\theta\) should be in radian measure and not in degree.

Step 3: Substitution of the Given Values

Insert the given radius, \(r = 16\) inches, and the central angle in radian (calculated in step 1) into the arc length formula to find the length of the arc.

Step 4: Rounding the Answer to Two Decimal Places

Finally, the resultant arc's length can be rounded to two decimal places as per the question requirements.

Key Concepts

Circle Geometry

Circle geometry deals with the study of the properties and measurements related to circles. A circle is a simple shape in geometry, where all points are equidistant from a central point called the center.

One of the main components of circle geometry is the radius—the distance from the center of the circle to any point on its circumference. In our exercise, the radius is given as 16 inches. The circumference of a circle is the total length around it, and this is calculated using the formula:

• Circumference = \(2\pi r\), where \(r\) is the radius.

Additionally, circles have angles known as central angles. These angles are formed by two radii coming out from the center, and they help define segments on the circle, such as arcs. Arcs are crucial because they mark part of the circle's circumference. Understanding these components is essential for calculations related to arc length, such as in our exercise.

Radian Conversion

When dealing with angles in circle geometry, it's often necessary to convert degrees into radians for mathematical calculations. A radian is another way to measure angles, and it conveys the extent of the angle based on the radius of the circle.

It's important to know the conversion factor:

• \(360^{\circ} = 2\pi\) radians.
• Thus, \(1^{\circ} = \frac{2\pi}{360} = \frac{\pi}{180}\) radians.

To convert an angle from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\). In our exercise, the central angle \(\theta\) is \(60^{\circ}\), which needs to be converted to radians as follows:

• \(\theta_{radians} = 60 \times \frac{\pi}{180} = \frac{\pi}{3}\) radians.

This conversion is a crucial step for calculating arc length, as the formula uses radians.

Central Angle Calculation

The central angle calculation is integral in finding the length of an arc on a circle. The central angle is the angle subtended at the center of the circle by the arc.

The length of an arc can be found using the formula:

• Arc length = \(r \times \theta\)

where \(r\) is the radius, and \(\theta\) is the central angle in radians. In the context of our exercise:

• The radius \(r\) is 16 inches.
• The central angle \(\theta\), converted to radians, is \(\frac{\pi}{3}\).

Substituting these values into the formula gives us:

• Arc length = \(16 \times \frac{\pi}{3} = \frac{16\pi}{3}\).

Expressing the arc length in terms of \(\pi\) and rounding it accordingly will give the desired measurement needed for this type of problem.