Problem 21 - Algebra for College Students

Question

$$ C=2 \pi r \quad \text { for } r $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The given equation is the formula for the circumference of a circle, where \( C \) represents the circumference and \( r \) represents the radius. We are required to solve for \( r \), meaning we need to express \( r \) in terms of \( C \).

Step 2: Isolate \( r \) on One Side of the Equation

To solve for \( r \), start by isolating \( r \) from the equation, \( C = 2 \pi r \). Divide both sides of the equation by \( 2 \pi \): \[ r = \frac{C}{2 \pi} \] Now, \( r \) is expressed in terms of \( C \).

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around the outer edge of the circle. Imagine you are wrapping a piece of string around a circular object like a coin. The length of the string that fits perfectly around the circle is the circumference. This is an important measure because it helps us understand the size of the circle's boundary.

The formula to calculate the circumference is:

• \( C = 2 \pi r \)

In this formula, \( C \) stands for the circumference, \( r \) is the radius of the circle, and \( \pi \) (pi) is a constant approximately equal to 3.14159. This formula is useful in a variety of applications, from designing tires to calculating the area of circular fields. Knowing the circumference helps in many practical situations where the size of a circle's path or boundary needs to be known.

Radius

The radius is a fundamental part of a circle. It is the length of a line from the center of the circle to any point on its edge. Think of it like the spoke of a bicycle wheel. Each spoke is a radius, extending from the hub (center) to the rim (edge).

In mathematics, the radius is often denoted by the letter \( r \), and it plays a crucial role in many formulas related to circles. The length of the radius directly influences other circle measurements, such as the circumference and the area. For example, when the radius doubles, the circumference also doubles. Therefore, understanding the concept of the radius is fundamental in geometry and is a building block for more complicated calculations involving circles.

Formulas in Mathematics

In mathematics, formulas are like recipes. They provide a step-by-step guide to solving problems or performing calculations. The use of formulas in mathematics allows us to find solutions efficiently and accurately. They often involve variables (like \( r \) for radius or \( C \) for circumference) and constants (like \( \pi \)).

For example, the formula to find the circumference of a circle, \( C = 2 \pi r \), uses these concepts by multiplying the constant \( 2 \pi \) with the variable \( r \). By substituting different values of \( r \), you can calculate the circumference for any circle.

• Formulas help simplify complex problems into manageable computations.
• They provide consistency and reliability across mathematical problems.

Learning how to use and manipulate formulas is a critical skill in mathematics and many scientific disciplines. It enables precise calculations and allows for the application of mathematical concepts to real-world scenarios, making problem-solving more approachable and clear. By mastering formulas, students can tackle a wide range of mathematical problems with confidence.