Problem 2 - Precalculus : Building Concepts and Connections

Question

Find the circumference of each circle given its radius or diameter. Leave your answer in terms of \(\pi .\) radius 5 inches

Step-by-Step Solution

Answer

The circumference of the circle is \(10\pi\) inches.

Step 1: Identify the given radius

From the problem, we know that the radius of the circle is 5 inches.

Step 2: Use the formula and substitute the radius

The formula for calculating the circumference of a circle is \(2\pi r.\) Substitute \(r\) with 5 inches into the formula: \(2\pi * 5 inches\).

Step 3: Simplify the expression to give the final answer

After multiplying, we get the circumference of the circle equals to \(10\pi\) inches

Key Concepts

Radius of a Circle

The radius of a circle is a straight line from the center of the circle to any point on its circumference. It is one of the most significant measurements in circle geometry, defining the size and the extent of the circle. Every point on the circumference is equidistant from the center, which illustrates the consistency of the radius throughout the circle. Understanding the concept of the radius is crucial as it directly affects other calculations related to the circle, such as area and circumference.

In practical questions, the radius can be given directly, or it can be deduced if the diameter is known, considering that the diameter is twice the radius. In the exercise provided, the radius is given as 5 inches, which would be the fundamental value used in computing the circumference using the relevant formula.

Circumference Formula

The circumference of a circle is the total distance around the circle, akin to the perimeter of a polygon. To calculate the circumference, you require the radius and a mathematical constant, \(\pi\). The formula to calculate the circumference, as applied in the exercise, is given by \(C = 2\pi r\), where \(C\) represents the circumference and \(r\) represents the radius of the circle.

The formula is a straightforward representation of the effect the radius has on the size of the circle. For the exercise in question, substiBUSINESS the radius (5 inches) into the formula results in an expression \(2\pi * 5 inches\), which simplifies to \(10\pi inches\). This simplified expression represents the circumference in terms of \(\pi\), which is often preferred for exact mathematical calculations.

Mathematical Constants

Mathematical constants are numbers with fixed values that arise naturally in various fields of mathematics. One such constant, central to circle geometry, is \(\pi\). The constant \(\pi\) represents the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. However, it is an irrational number, meaning it has an infinite number of non-repeating decimals.

Using \(\pi\) in formulas allows mathematicians and students to calculate the circumference (or area) of a circle with great precision. In our exercise, \(\pi\) is kept in the solution to maintain accuracy. Notably, \(\pi\) is not the only mathematical constant, but it is specifically essential when dealing with circle-related problems. Real-world applications often use the approximation \(3.14\) or \(\frac{22}{7}\), but exact calculations require retaining the symbol \(\pi\) within the answer.