Problem 94 - College Algebra and Calculus: An Applied Approach

Question

Solve for the indicated variable. Area of a Sector of a Circle Solve for \(\theta\) in \(A=\frac{\pi r^{2} \theta}{360}\).

Step-by-Step Solution

Answer

The angle \(\theta\) in degrees can be represented as \(\theta = \frac{360A}{\pi r^{2}}\), given the area of the sector \(A\), and the radius of the circle \(r\).

Step 1: Isolate \(\theta\)

First, isolate the term with \(\theta\). This can be achieved by multiplying both sides of the equation by \(\frac{360}{\pi r^{2}}\) to get \(\theta = \frac{360A}{\pi r^{2}}\).

Step 2: Simplify the Equation

The equation is \(\theta = \frac{360A}{\pi r^{2}}\), which represents \(\theta\) in terms of \(A\), \(r\), and known constants (360 and \(\pi\)). The formula is now simplified and solved for \(\theta\).

Key Concepts

Area of a Sector

To understand the area of a sector, imagine slicing a pizza into several equal pieces. Each slice represents a sector of the pizza, which is a part of a circle. When addressing the area of a sector, we're finding the size of the 'pizza slice' within the whole 'pizza pie' or circle. The area is proportional to the angle formed between the two radii, known as the central angle, and the formula used for this is broadly stated as sector area (\(A\)) = \frac{1}{2} * \text{radius}^2 * \text{central angle} (in radians).
For our purposes, we use this formula with the central angle in degrees:

\[ A = \frac{\pi r^{2} \theta}{360} \]
In this formula,

• \(r\) stands for the radius of the circle,
• \(\theta\) is the central angle in degrees,
• \(A\) is the area of the sector, and
• \(\pi\) is a constant approximately equal to 3.14159.

This equation allows you to find the area of any sector if you know the radius of the circle and the central angle. Conversely, if you know the area and either the radius or the angle, you can solve for the other missing variable.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations or expressions to reveal the desired information or solve for a particular variable. It's a fundamental skill in mathematics, and mastering it can help one make sense of and solve complex problems.

In the context of solving geometry problems such as finding the area of a sector, algebra comes into play when we work to isolate a specific variable. Techniques involve

• multiplying or dividing both sides of an equation by the same number,
• adding or subtracting the same number from both sides, and
• using inverse operations to undo addition, subtraction, multiplication, or division.

Applying algebraic manipulation effectively requires an understanding of which operations to use and in what sequence to apply them to isolate the variable in question.

Circle Geometry

Circle geometry encompasses a variety of concepts related to the properties and measurements of circles. Understanding that a circle is a shape with all points equidistant from a center point is just the beginning. Circle geometry involves several components such as radius, diameter (twice the radius), circumference (the perimeter of the circle), and area.

Central Angles and Arcs

A central angle is formed by two radii and determines the size of the arc, which is the portion of the circle's circumference covered by the angle. The relationship between the circle's area, sectors, and central angles is critical to solving many circle geometry problems, such as finding the area of a sector. In equations,

• the radius is often noted as \(r\),
• the circumference as \(C\),
• and the central angle in degrees as \(\theta\).

When we speak of 'solving for variables in geometry', we're often finding the value of one of these components given the others.

Isolating Variables

Isolating a variable means to rewrite an equation so that a particular variable is alone on one side of the equation, and everything else is on the other side. In the example provided where we solve for \(\theta\), the process of isolating the variable is crucial.
By performing algebraic operations that 'undo' the relationship between \(\theta\) and the other variables in the equation, we can find \(\theta\) as a single value. This process often involves inverse operations such as division if the variable is multiplied, or addition if it is being subtracted, among others. The goal is to reverse the operations affecting the variable, step by step, until it stands alone. In our case, this is achieved by the step-by-step process, ultimately resulting in the expression \[ \theta = \frac{360A}{\pi r^{2}} \], which shows \(\theta\) with respect to \(A\) and \(r\), isolated and ready to be calculated.