Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Find the length of each arc intercepted by a central angle \(\boldsymbol{\theta}\) in a circle of radius \(\kappa\) Round answers to the nearest hundredth. $$r=71.9 \text { centimeters; } \theta=135^{\circ}$$

Step-by-Step Solution

Answer

The arc length is approximately 169.08 centimeters.

Step 1: Convert Degrees to Radians

The formula to convert the angle from degrees to radians is \( \text{radians} = \theta \times \frac{\pi}{180} \). For \( \theta = 135^{\circ} \), we have:\[ 135 \times \frac{\pi}{180} = \frac{3\pi}{4} \text{ radians.} \]

Step 2: Calculate Arc Length Using the Formula

The arc length \( L \) is given by the formula \( L = r\theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. Using \( r = 71.9 \) centimeters and \( \theta = \frac{3\pi}{4} \),\[ L = 71.9 \times \frac{3\pi}{4} \approx 169.0778573 \text{ centimeters.} \]

Step 3: Round the Result to Nearest Hundredth

We need to round the previously calculated arc length to the nearest hundredth. \( 169.0778573 \) rounds to \( 169.08 \) when rounded to two decimal places.

Key Concepts

Radians

Radians are a unit of angular measurement used in mathematics, and especially useful in circle geometry. Unlike degrees, which divide a circle into 360 parts, radians relate the arc length of a circle to its radius. This makes them particularly valuable for calculations involving circles.
A full circle in radians is denoted as \( 2\pi \), because the circumference of a circle is \( 2\pi r \). Each radian measures the angle created when the arc length is equal to the radius. Therefore, the conversion from degrees to radians is given by the formula:

• \( \text{radians} = \theta \times \frac{\pi}{180} \)

For example, an angle of \( 135^{\circ} \) can be converted to radians by multiplying by \( \frac{\pi}{180} \), resulting in \( \frac{3\pi}{4} \) radians.
Understanding radians is crucial when working with trigonometric functions, calculus, and geometry, as they provide a consistent and simple way to deal with angular measurements in mathematical equations.

Central Angle

The central angle is a key term in circle geometry. It is the angle formed by two radii that extend from the center of a circle to its circumference. The size of this angle relates directly to the arc length that it intercepts on the circle.
The formula for finding the arc length when the central angle \( \theta \) is measured in radians is:

• \( L = r\theta \)

Here, \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. This is why converting angles from degrees to radians is often necessary when calculating arc length. For example, with a radius of 71.9 centimeters and a central angle of \( \frac{3\pi}{4} \) radians, the arc length is calculated as \( L = 71.9 \times \frac{3\pi}{4} \approx 169.08 \) centimeters.
Central angles are fundamental in many applications, including geometry, navigation, and even designing gears and wheels, as they determine the span of the circle segment involved.

Circle Geometry

Circle geometry is a branch of mathematics focusing on the properties and figures surrounding circles. It involves various elements such as radii, diameters, chords, tangents, and arcs, providing a rich field for learning fundamental geometric concepts.
At the heart of circle geometry is the relationship between angles and arcs. The central angle theorem, for example, connects an arc's angle to the central angle, as discussed previously. Understanding how these properties interact allows for precise calculations and predictions about circular shapes and their segments.
Furthermore, the concepts of circumference and area crucially depend on the circle's radius. These formulas, \( C = 2\pi r \) for circumference and \( A = \pi r^2 \) for area, highlight the circle's symmetry and constant \( \pi \), linking all aspects of the circle geometry.
Whether it's used in real-life scenarios such as engineering or theoretical problems in mathematics, mastering circle geometry equips learners with tools to solve complex problems involving rounded shapes, such as determining distances, optimizing materials, and understanding rotational motion.