Problem 1 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(1-4,\) the angle lies at the center of a circle and subtends an arc of the circle. Find the missing angle measure, circle radius, or arc length. \(\begin{array}{ll}{\text { Angle }} & {\text { Radius }} &{\text {Arc Length }} \\ {5 \pi / 8} & {2} & {? }\end{array}\)

Step-by-Step Solution

Answer

The arc length \(s\) is \( \pi 5/4 units\).

Step 1: Identify Given Information

The angle \(\theta\)= \(5 \pi / 8\) rad and the radius \(r\)=2.

Step 2: Apply the Arc Length Formula

The arc length \(s\) can be found by using the formula: \(s = r \times \theta\). Hence, substituting the values: \(s = 2 \times (5 \pi / 8) \).

Step 3: Simplify Expression

Calculate the multiplication: \(s= \pi 5/4\). This will give the length of arc subtended by the given angle at the center of the circle.

Key Concepts

Angle Measure

The concept of angle measure is critical in understanding arcs and circles. An angle is often measured in radians, particularly in the context of circles. Radians provide a measure that relates the angle directly to the circle's radius. In this exercise, the angle \(\theta\) given is \(\frac{5\pi}{8}\) radians. But what does that mean?
This angle measure implies the extent of rotation from one radius to another that subtends an arc on the circle's circumference. The radian measure is the length of the arc divided by the radius of the circle. So, in simple words:

• 1 radian is the angle created when the radius and the arc length are equivalent.
• \(\pi\) radians equal 180 degrees, meaning that in one full circle, there are \(2\pi\) radians.

Understanding this helps with calculations involving aspects of a circle such as arc length and sector area.

Circle Radius

The radius of a circle is a fundamental concept that influences many calculations involving circles, including the arc length and area. A circle's radius is the distance from its center to any point on its circumference. In this exercise, the radius \(r\) is given as 2.
Why is the radius important? Because it serves as a scaling factor when determining the size of sections of the circle, like arcs and sectors.

• The radius is always constant for a particular circle, making it a reliable measure for calculations.
• It is crucial in the formula for arc length: \(s = r \times \theta\), where \(s\) is the arc length.
• It's also used to calculate other circle properties, such as the area (\(\pi \cdot r^2\)) and the circumference (\(2 \cdot \pi \cdot r\)).

Understanding the circle radius allows us to link angles and arc lengths effectively.

Arc Length Formula

The arc length formula is a pivotal tool in circle geometry, particularly when working with central angles. To find the arc length \(s\), we use the formula: \(s = r \times \theta\), where \(r\) is the circle radius and \(\theta\) is the angle in radians.
This formula is essential because it connects three important circle properties, providing a means to resolve one when the other two are known:

• The formula shows that the arc length is directly proportional to both the radius and the angle.
• By changing the angle or radius, the arc length adjusts accordingly, which helps to model real-world situations accurately.
• For example, a larger angle or radius increases the arc length, illustrating the relationship between these variables.

Using this formula, as seen in the exercise, clarifies how angle measures and radii combine to form arcs of specific lengths.