Problem 8 - Basic Engineering Mathematics

Question

Find the length of arc of a circle of radius \(5.5 \mathrm{~cm}\) when the angle subtended at the centre is \(1.20\) radians.

Step-by-Step Solution

Answer

The arc length is 6.6 cm.

Step 1: Understand the problem

We need to find the length of an arc of a circle. The circle has a radius of \(5.5\) cm, and the angle subtended by the arc at the center of the circle is \(1.20\) radians.

Step 2: Use the formula for arc length

The formula for the length of an arc \(L\) of a circle is \(L = \theta \times r\), where \(\theta\) is the angle in radians and \(r\) is the radius of the circle.

Step 3: Substitute the given values into the formula

Substitute \(\theta = 1.20\) and \(r = 5.5\) cm into the formula, giving \(L = 1.20 \times 5.5\).

Step 4: Calculate the arc length

Calculate \(L = 1.20 \times 5.5 = 6.6\) cm.

Key Concepts

Circle Geometry

When studying circle geometry, we encounter various elements that make up a circle. From the center to the circumference, each piece has its significance. The fundamental components in this topic involve identifying the circle's radius, diameter, and the key angles that are formed within the circle.

- **Radius**: This is the line segment that extends from the center of the circle to any point on the perimeter. In our example, the radius is given as 5.5 cm. - **Diameter**: It is the longest line that passes through the center of the circle, effectively doubling the radius. However, for this problem, we focus on the radius only. - **Angle at the Center**: It is the angle subtended by two radii forming an arc on the circle's circumference. Understanding these elements is crucial for solving problems related to arcs and sectors in circle geometry. Knowing how to manipulate these parts allows us to calculate various circle attributes like the arc length, area of sectors, and more.

In the given problem, we focus specifically on the relationship between the radius and the angle to find the arc length.

Radian Measure

Radian measure is a way of expressing angles relative to the radius of a circle. Unlike degrees, radians provide a more natural way to look at the angle because they relate directly to the circle’s geometry. - **Radian Definition**: A radian is the angle created when the radius lays off an arc on the circle whose length equals the radius itself. So, it ties the angle directly to the concept of circumference. - **Conversion**: While degrees are more commonly used in everyday contexts (1 full circle = 360°), radians (1 full circle = 2π radians) provide a simpler, more direct approach in mathematics and physics. In applications involving the calculation of arc length, radian measure is especially handy because it allows the arc length formula to be represented as a simple multiplication, eliminating unnecessary conversion steps. In our exercise, the angle is given as 1.20 radians, which simplifies the calculation of arc length using the formula discussed below. Understanding radians helps you see how angles naturally fit into the circle, making calculations smoother and more intuitive, especially in calculus and advanced geometry.

Arc Length Formula

The arc length formula is a key concept in circle geometry. It allows you to find the distance along the curved path of a circle's arc. This is particularly useful in engineering, architecture, and even art. The formula is expressed as: \[ L = \theta \times r \]where:- **\(L\)**: Arc length - **\(\theta\)**: Angle in radians- **\(r\)**: Radius of the circle To find the arc length, you simply multiply the radius of the circle by the radian measure of the angle subtending at the circle's center. In our problem, we have:\[ L = 1.20 \times 5.5 = 6.6 \text{ cm} \]This formula is intuitive when using radians because one radian is directly tied to the radius and the length of the arc equivalent to it.By understanding and applying the arc length formula, you can easily determine the length of any arc, provided you know the angle in radians and the radius of the circle. This section wraps up the exercise, showing how straightforward it can be to determine distances on a circle's perimeter when using the appropriate tools.