Problem 51 - Algebra A Combined Function

Question

Solve each equation for the indicated variable. $$ \frac{C}{\pi r}=2 \text { for } r \text { (Geometry: circumference of a circle) } $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is derived from the formula for the circumference of a circle. The equation is \( \frac{C}{\pi r} = 2 \). We need to solve for \( r \).

Step 2: Eliminate the Fraction

Multiply both sides of the equation by \( \pi r \) to eliminate the fraction. This gives: \( C = 2 \pi r \).

Step 3: Solve for \( r \)

To isolate \( r \), divide both sides by \( 2\pi \) to get: \( r = \frac{C}{2\pi} \).

Key Concepts

Circumference of a Circle

The circumference of a circle is essentially the distance around the circle, which can be considered as the perimeter of the circle. Understanding this concept is very helpful when dealing with circle-related equations. To calculate the circumference, you use the formula \( C = 2\pi r \), where \( C \) stands for circumference, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

• The circumference relates to the radius, which is the distance from the center of the circle to its outer edge.
• \( \pi \) is a crucial element of the equation, reflecting the unique ratio of a circle's circumference to its diameter.

In problems like the one provided, this equation can often be manipulated to solve for different variables, such as the radius or even \( \pi \) itself. Mastering how to use and rearrange the circumference formula is key for many geometry and algebra problems.

Isolation of Variables

Isolation of variables involves rearranging an equation so that you solve for one particular variable. This is an essential skill in algebra as it allows you to find the unknown quantity in an equation. Here is how one would apply it to the given problem:To isolate the variable, you need to perform opposite operations to "free" the variable on one side of the equation while keeping the equation balanced:

• In the original equation \( \frac{C}{\pi r} = 2 \), the goal is to isolate \( r \).
• You can begin by eliminating any fractions, multiples, or additional terms that encompass the variable you need.
• The next step is to make sure \( r \) stands by itself, which eventually shows the readers how each step logically arrives at \( r = \frac{C}{2\pi} \).

You do this methodically, ensuring every move is mathematically correct. Practice is critical as this skill helps in solving more complex problems too.

Eliminating Fractions

Fractions can make equations seem more complicated but eliminating them makes equations easier to handle. In the given mathematical problem, eliminating the fraction is one of the first steps:

• The original equation is \( \frac{C}{\pi r} = 2 \).
• To remove the fraction, multiply every term by the denominator \( \pi r \). This action gets rid of the fraction by creating a balanced equation on both sides.
• You'll then have \( C = 2\pi r \), a simpler equation to work with without fractions.

Remember, moving terms to eliminate fractions is a common strategy not just in geometry but across all algebra. It transforms a seemingly complex problem into something far more approachable. Mastering this step allows you to handle more advanced algebraic manipulations with confidence.