Problem 63 - Precalculus Essentials

Question

A circle has a radius of 8 inches. Find the length of the arc intercepted by a central angle of \(150^{\circ} .\) Express are length in terms of \(\pi .\) Then round your answer to two decimal places. (Section 4.1, Example 8)

Step-by-Step Solution

Answer

The length of the arc is \( \frac{5}{3} \pi \) inches or approximately \( 5.24 \) inches when rounded to two decimal places.

Step 1: Identify Knowns

At this point, take note of the known quantities from the exercise: the radius \( r = 8 \) inches and the central angle \( \theta = 150 \) degrees.

Step 2: Use Proportion to Calculate Arc Length

For a complete circle, the arc length is equivalent to its circumference, which is given by \( 2\pi r \), and the central angle is \( 360 \) degrees. Suppose \( L \) is the arc length for our central angle, then we establish the proportion \(\frac{L}{2\pi r} = \frac{\theta}{360} \). Solving for \( L \), we get \( L = \frac{\theta}{360} \times 2\pi r \). Substituting the known values, we get \( L = \frac{150}{360} \times 2 \times \pi \times 8 = \frac{5}{3} \pi \) inches.

Step 3: Round your Answer

Perform the multiplication and round your answer to two decimal places. This is approximately \( 5.24 \) inches.

Key Concepts

Central Angle

The concept of a central angle is vital when it comes to understanding various properties and calculations related to circles. A central angle is defined as the angle whose vertex is at the center of the circle and whose sides (or legs) extend to the circumference. This angle essentially 'opens up' to 'embrace' an arc on the circle. The size of a central angle in a circle will determine the length of the arc it intercepts.

As seen in our exercise, the central angle was given as 150 degrees, which is less than a full circle (360 degrees). This proportionality is the key to unlocking the length of the corresponding arc. By comparing the ratio of the central angle to the full angle of a circle, one can determine how 'big' or 'small' the arc is relative to the full circumference. It's like slicing a pizza—the angle of your slice determines how large your piece will be! Knowing how to work with central angles in both degrees and radians is crucial for problems involving arc length, sector area, and various applications in geometry and trigonometry.

Circle Circumference

Circle circumference represents the total distance around a circle, which is directly proportional to its radius. The formula to calculate the circumference, as referenced in the above exercise, is \(2\pi r\), where \(r\) is the radius. Circumference is like the ‘perimeter’ for a circle, and it plays a pivotal role in calculating the length of an arc.

To understand the relation between arc length and circumference, imagine a string being wrapped perfectly around the outer edge of a circular object. The length of that string is the circumference. When you're looking at only a portion of the circle, as delimited by the central angle, you are calculating a fraction of this string's length, which is the arc length. In our problem, after we’ve determined the portion of the circle our central angle covers, we apply the same portion to the full circumference to find our arc length. It's like measuring a part of the string that matches the angle of our 'slice' of the circle.

Radians and Degrees

Radians and degrees are two units of measurement used to quantify the size of angles. Degrees are denoted by the symbol ° and are the more commonly used unit in everyday applications. One full rotation around a circle is 360°. On the other hand, radians offer a more natural way of measuring angles based on the radius of the circle, and are often used in higher mathematics and sciences. There are \(2\pi\) radians in one complete revolution, which means that \(360° = 2\pi \text{ radians}\).

To convert from degrees to radians, you can use the conversion factor \(\pi/180°\). Conversely, to convert from radians to degrees, you multiply by \(180°/\pi\). The exercise could also have asked for the arc length in radians, requiring a different but equally straightforward approach. You would then use the arc length formula \(L = r\theta\), with \(\theta\) in radians, to arrive at the answer. Understanding how to switch between radians and degrees is important because it allows flexibility and is required in different contexts of mathematics and physics.