Problem 45 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The circumference of a circle measures \(50 \pi\) units. Find the radius.

Step-by-Step Solution

Answer

The radius is 25 units.

Step 1: Understand the Formula for Circumference

The formula for the circumference of a circle is given by \( C = 2\pi r \), where \( C \) is the circumference and \( r \) is the radius.

Step 2: Set up the Equation

We are given that the circumference \( C = 50\pi \). Substitute this value into the formula: \( 50\pi = 2\pi r \).

Step 3: Simplify the Equation

Since both sides of the equation include \( \pi \), we can divide both sides by \( \pi \) to simplify the equation: \( 50 = 2r \).

Step 4: Solve for the Radius

To find \( r \), divide both sides of the equation by 2: \( r = \frac{50}{2} \).

Step 5: Calculate the Solution

Compute \( \frac{50}{2} \) to find \( r \). The result is \( r = 25 \).

Key Concepts

Circle Circumference

When dealing with circles, one of the most fundamental concepts is the circle's circumference. The circumference is the distance around the circle, comparable to the perimeter of a polygon. Understanding the equation for the circumference is vital:

• The standard formula for circumference is \( C = 2\pi r \), where \( C \) represents the circumference, \( \pi \) is approximately 3.14159, and \( r \) is the radius of the circle.
• This formula shows the relationship between the length around the circle (circumference) and how far it stretches directly from the center (radius).
• \( \pi \) is a constant that represents the ratio of any circle's circumference to its diameter. It's essential when working with circles.

Breaking down the formula reveals how each piece works together to describe the circle's most basic properties. In exercises involving the circumference, you often solve for other variables like the radius, given the circumference value.

Solving Equations

Solving equations is a fundamental skill in algebra that allows us to find unknown values. In this context, we use algebraic techniques to solve for the radius of a circle. Here’s a step-by-step approach:

• Begin with the known formula: \( 50\pi = 2\pi r \).
• Notice that \( \pi \) appears on both sides of the equation. You can divide both sides by \( \pi \) to reduce complexity, leading to \( 50 = 2r \).
• This process is called simplification. It clears up the equation by removing the same factor from both sides.
• Next, to isolate \( r \), divide both sides by 2. This step is crucial as it lets us solve for the radius: \( r = \frac{50}{2} \).
• Performing the arithmetic gives \( r = 25 \).

Each step follows logically from the previous one, maintaining the balance of the equation while achieving the goal of isolating the desired variable.

Radius Calculation

The radius is a key measure in circles, describing the distance from the circle's center to its edge. Calculating the radius is straightforward once the circumference is known:

• Using the equation \( C = 2 \pi r \), you solve for \( r \) by rearranging the equation: \( r = \frac{C}{2\pi} \).
• In the solved equation \( r = \frac{50}{2} \), this can be interpreted as dividing the given circumference by the factor \( 2\pi \).
• The division by 2 ensures that we are focusing solely on half of the diameter, aligning directly with the radius definition.
• The step-by-step solving of the equation not only delivers the correct radius value but also reinforces the understanding of how each part of the circumference formula connects to this critical geometrical concept.

Thus, the radius calculation illustrates why algebraic manipulation and clear understanding of formulas are essential in solving geometry-related problems.