Problem 12 - Technical Mathematics with Calculus

Question

In the following exercises, \(s\) is the length of are subtended by a central angle \(\theta\) in a circle of radius \(r\). \(\theta=2.08 \mathrm{rad}, s=3.84 \mathrm{m} . \quad\) Find \(r\)

Step-by-Step Solution

Answer

The radius of the circle is 1.84615 meters.

Step 1: Recall the formula relating arc length, radius, and central angle in radians

The relationship between the arc length, radius, and central angle in radians is given by the formula: \( s = r * \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.

Step 2: Rearrange the formula to solve for the radius

To find the radius \( r \), rearrange the formula to solve for \( r \): \( r = \frac{s}{\theta} \).

Step 3: Plug in the values for \( s \) and \( \theta \) and solve for \( r \)

Substitute the given values \( s = 3.84 \) meters and \( \theta = 2.08 \) radians into the rearranged formula: \( r = \frac{3.84}{2.08} \). Now calculate the value of \( r \).

Step 4: Calculate the radius

Perform the division to find the radius: \( r = \frac{3.84}{2.08} = 1.84615 \) meters.

Key Concepts

Central Angle in Radians

Understanding the concept of a central angle measured in radians is crucial when dealing with circles. A central angle is formed by two radii of a circle and the curve of the circle between those lines. In simpler terms, it's the angle formed at the center of the circle that subtends, or 'stands under', an arc on the circle's edge.

Radians are a way of measuring angles based on the radius of the circle. Every circle, regardless of size, has 2π radians, corresponding to the circle's circumference. There's a direct relationship between the measure of the central angle in radians and the length of the arc it intercepts. Each radian measures approximately 57.2958 degrees, making it a more natural unit for angles in mathematics, particularly calculus and trigonometry, due to its direct connection to the circle's properties.

Circle Radius

The radius of a circle is the distance from the center of the circle to any point on its circumference. It's a fundamental property of a circle and is used in numerous formulas to calculate various attributes of the circle, such as its area, circumference, and, as we've seen, the arc length.

When visualizing radius, picture it as a line segment that connects the center of the circle with the circle's edge. All these line segments, regardless of direction, will have the same length, and this uniform length is a defining feature of the circle. Furthermore, the radius of a circle forms half of its diameter, which is the longest distance between two points on the circle’s circumference, passing through the center.

Relationship Between Arc Length and Radius

The relationship between arc length and the circle's radius is elegantly simple, yet deeply fundamental in geometry. This relationship is encapsulated by the formula: \( s = r \theta \), where \( s \) represents the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.

The formula shows us that the arc length is directly proportional to the radius for a fixed central angle. This means that larger circles will have longer arcs than smaller circles if the central angles are the same. Conversely, for a given circle, as the central angle increases, the length of the corresponding arc also increases. This proportionality is why radians are such a natural way to measure angles for circular motion and phenomena.