Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

It is possible to have a circle whose circumference is numerically equal to its area.

Step-by-Step Solution

Answer

The radius of a circle required for its area and circumference to be numerically equal is 2 units.

Step 1: Formulate the equation

Equating the formulas for circumference and area of a circle gives: \(2\pi r = \pi r^2\).

Step 2: Simplify the equation

Simplify the equation to solve for \(r\). First, divide both sides by \(\pi r\), to give \(2 = r\).

Step 3: Solve for \(r\)

Since \(r = 2\) equates the formulas for the area and circumference of a circle, it is the the radius needed for the circle to have numerically equal circumference and area.

Key Concepts

Circumference

The circumference of a circle is the distance around the circle. It essentially measures the "outline" you see when picturing a circle in your mind. To find the circumference, you use the formula:

\[C = 2\pi r\]This formula involves two key components:

• \(C\): The circumference itself.
• \(r\): The radius, which is the distance from the center of the circle to any point on its edge.

The Greek letter \(\pi\) is approximately 3.14159, and it represents the ratio of the circumference of any circle to its diameter. That is why it is involved in circle calculations.

To understand how this works, imagine a circular track. If you were to walk around it, the distance you’d travel is the circumference. Knowing the radius helps you calculate how long that walk would be, no matter how big or small the track is.

Area

The area of a circle measures the space it occupies within its boundaries. For a clearer analogy, it's like painting a circular disc entirely; the area represents how much paint you would need.

The formula used to calculate the area is:

• \(A = \pi r^2\)

Where:

• \(A\) is the area of the circle.
• \(r\) is, once again, the radius.

Calculating the area can help in many practical tasks, from determining how much material is needed to make a circular tablecloth to figuring out the size of a circular field. The squared term \(r^2\) makes it essential to square the radius first, then multiply by \(\pi\). This is what makes the unit for area square units, such as square meters or square inches.

Equation solving

Solving equations requires a clear understanding of balancing elements to find unknown values. In our context, we deal with an equation where we equate the circumference and area of a circle:
The first step is to use their respective formulas:

• \(2 \pi r = \pi r^2\)

In troubleshooting how to handle this equation:

1. **Equate and Simplify**: We start by eliminating \(\pi\) from both sides, assuming \(\pi\) cannot be zero.

• This simplifies the equation to \(2 = r\).

2. **Solve for \(r\)**: At this point, you've isolated \(r\) as our unknown. To finalize, we've concluded that the radius \(r\) must be 2 for the circle's circumference to equal its area.

Recognizing scenarios where this equation holds true, like the given solution, requires careful simplification and understanding of basic algebraic principles. Whether in math or practical applications, equation solving gives us solutions to real-world problems by representing them mathematically.