Problem 74 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } r: \quad C=2 \pi r $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

We are given the equation \( C = 2\pi r \) and need to solve for the variable \( r \). This means we need to isolate \( r \) on one side of the equation.

Step 2: Identify the Operation with r

The variable \( r \) is being multiplied by \( 2\pi \). To isolate \( r \), we need to undo this multiplication.

Step 3: Divide Both Sides by 2π

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

Step 4: Final Expression for r

The rearranged formula shows that \( r = \frac{C}{2\pi} \). This is the expression for \( r \) when \( C \) is the given circumference.

Key Concepts

Variable Isolation

When solving equations, "variable isolation" is a key skill to master. The aim is to get the variable you want to solve for on one side of the equation and everything else on the other side. Let's break it down further:

• Identify the variable you need to isolate.
• Determine the operations being applied to that variable.
• Use inverse operations to cancel out these operations one by one.

For instance, in the equation \( C = 2\pi r \), our goal is to isolate \( r \). We see that \( r \) is multiplied by \( 2\pi \). To isolate \( r \), you must perform the inverse operation, which is division by \( 2\pi \). By following these steps, you effectively separate \( r \) from the rest of the equation.

Division in Equations

Division is often used in equations to simplify and solve for a variable. It is an inverse operation and is particularly useful when a variable is being multiplied by a coefficient. Here's how you can use division to solve an equation:

• Identify the multiplication with the variable.
• Divide both sides of the equation by the same non-zero number to keep the equation balanced.

In our exercise, \( C = 2\pi r \) represents a situation where \( r \) is multiplied by \( 2\pi \). To isolate \( r \), we must divide both sides by \( 2\pi \):\[ r = \frac{C}{2\pi} \]This step allows us to solve for \( r \), effectively expressing \( r \) in terms of \( C \). Remember, whatever operation you do to one side, you must do to the other to maintain equality.

Circumference Formula

The circumference formula \( C = 2\pi r \) is fundamental in geometry, especially when dealing with circles. This formula expresses the total distance around a circle, or its perimeter, using the circle's radius.

• "\( C \)" stands for the circumference.
• "\( 2\pi \)" is a constant, roughly equivalent to 6.283, accounting for the circle's diameter relation to its radius.
• "\( r \)" is the radius, the distance from the circle's center to any point on the perimeter.

To solve for \( r \) when you know \( C \), rearrange the formula to \( r = \frac{C}{2\pi} \). Understanding this formula is essential in many branches of mathematics and real-world applications, such as determining the size of wheels, pipes, and circular objects.