Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(C=2 \pi r\) for \(r\)

Step-by-Step Solution

Answer

The radius \(r\) is \(\frac{C}{2\pi}\).

Step 1: Understand the Problem

The formula given is for the circumference of a circle, which is expressed as \(C = 2\pi r\). You need to solve this equation for the radius \(r\).

Step 2: Isolate the Variable

To solve for \(r\), you first need to isolate \(r\) on one side of the equation. You can do this by dividing both sides of the equation by \(2\pi\).

Step 3: Perform the Division

Divide both sides of the equation \(C = 2\pi r\) by \(2\pi\):\[\frac{C}{2\pi} = \frac{2\pi r}{2\pi}\]

Step 4: Simplify the Equation

On the right side, \(2\pi\) in the numerator and denominator will cancel out, leaving you with:\[\frac{C}{2\pi} = r\]

Step 5: Conclusion: Solved for the Variable

Now, the formula is rearranged to solve for \(r\): \[r = \frac{C}{2\pi}\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a vital skill in mathematics, allowing you to rearrange equations and solve for unknown variables. It's like reshaping a puzzle until all pieces fit perfectly. In our exercise, you begin with the formula for the circumference of a circle, \(C = 2\pi r\). The goal is to express this equation differently, making it easier to solve for the variable \(r\).
To achieve this, understanding the properties of equations is essential. Equations are like balanced scales; you must perform the same operations on both sides to maintain equality. Here, the operation involved is division, which helps to isolate \(r\) on one side. Algebraic manipulation is about knowing when and how to apply these operations effectively. This skill not only helps in solving math problems but also in understanding complex relationships between variables in real-world scenarios.

Circle Circumference

The circumference of a circle is the linear distance around its edge. Imagine walking around a circular track; your path's length represents the circle's circumference.
The mathematical formula for calculating the circumference is \(C = 2\pi r\). In this formula, \(C\) stands for circumference, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius of the circle. The term \(2\pi r\) highlights that the circumference is proportional to the radius. This means if you double the radius, the circumference also doubles.
This formula is foundational in geometry and helps solve various real-life problems involving circles, like determining the perimeters of cylindrical objects, or even understanding the orbit of planets!

Variable Isolation

Variable isolation involves rearranging an equation to express one variable in terms of others. Think of it as untangling a knot to see the individual strings clearly.
In the problem at hand, the task is to isolate the variable \(r\) from the equation \(C = 2\pi r\). You do this by dividing both sides of the equation by \(2\pi\), which "undoes" the multiplication of \(2\pi\) with \(r\).

• Start: \(C = 2\pi r\)
• Divide both sides: \(\frac{C}{2\pi} = \frac{2\pi r}{2\pi}\)
• Cancel out \(2\pi\): \(\frac{C}{2\pi} = r\)

Now, \(r = \frac{C}{2\pi}\) becomes isolated, clearly showing \(r\)'s value in terms of known quantities. This isolation process is crucial in mathematics, allowing for the clear and concise expression of relationships between variables. Effective variable isolation helps in solving problems across all scientific and engineering disciplines.