Problem 13 - Algebra for College Students

Question

Solve each formula for the specified variable. \(C=2 \pi r\) for \(r\) (circumference of a circle)

Step-by-Step Solution

Answer

r = \frac{C}{2 \pi}\

Step 1: Understand the given equation

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

Step 2: Isolate the variable \(r\)

To solve for \(r\), we need to get \(r\) alone on one side of the equation. We do this by dividing both sides of the equation by \(2 \pi\). This step can be written as: \[ r = \frac{C}{2 \pi} \]

Key Concepts

Circumference

The circumference of a circle refers to the total distance around the edge of the circle. In simpler terms, it's like the perimeter of the circle. The formula to calculate the circumference, when you know the radius, is given by \( C = 2 \pi r \). Here:

• \( C \) is the circumference.
• \( r \) is the radius of the circle.
• \( \pi \) (Pi) is a mathematical constant approximately equal to 3.14159.

The formula essentially tells us that multiplying the radius by 2 and then by Pi gives us the distance around the circle. This foundational formula is used often in geometry and various real-world applications.

Formula Manipulation

Formula manipulation is a fundamental concept in algebra and involves rearranging or adjusting formulas to solve for a specific variable. In our given problem, we start with the equation for circumference \( C = 2 \pi r \), and we need to find the radius \( r \).
To do this, we manipulate the formula to isolate \( r \). Here are the steps:

• Identify what you need to solve for, which in this case is \( r \).
• Rearrange the formula to get the variable \( r \) alone.
• Divide both sides of the equation by \( 2 \pi \).
• This eliminates the \( 2 \pi \) from the right side and leaves \( r \).

The new formula to find \( r \) when given \( C \) becomes \( r = \frac{C}{2 \pi} \). This process of changing the arrangement of the formula while maintaining equality is critical in algebra for solving various equations.

Algebraic Isolation

Algebraic isolation refers to the process of getting a specific variable alone on one side of an equation. This technique is vital for solving equations and is used in our exercise.
Let's go through the steps to isolate \( r \) in the circumference formula \( C = 2 \pi r \):

• First, recognize that you need \( r \) on its own.
• Notice that \( r \) is multiplied by \( 2 \pi \).
• To undo this multiplication, do the opposite operation: divide.
• Divide both sides of the equation by \( 2 \pi \): \( r = \frac{C}{2 \pi} \).

This method ensures \( r \) is by itself on one side of the equation, making it easy to determine the radius when the circumference is known. Mastering algebraic isolation helps in solving more complex problems by breaking them down into manageable steps.