Problem 63 - Introductory Algebra for College Students

Question

Solve the formula for the specified variable. Because each variable is nonnegative, list only the principal square root. If possible, simplify radicals or eliminate radicals from denominators. $$A=\pi r^{2} \text { for } r$$

Step-by-Step Solution

Answer

The solution to the equation \(A = \pi r^{2}\) for \(r\) is \(r = \sqrt{A/\pi}\)

Step 1: Understand the equation

The given equation is \(A = \pi r^{2}\), which represents the formula for the area of a circle with radius \(r\). The exercise involves solving for \(r\), meaning we need to isolate \(r\) on one side of the equation.

Step 2: Divide by pi

First, divide both sides of the equation by \(\pi\) to isolate \(r^{2}\): \(A/\pi = r^{2}\)

Step 3: Apply square root to both sides of the equation

Next, apply square root to both sides of the equation to get \(r\) alone. The square root of \(r^{2}\) is \(r\): \(\sqrt{A/\pi} = r\)

Step 4: Consideration of non-negative roots

Given that each variable is non-negative, only the principal square root (positive root) should be considered. Therefore, \(r\) equals to \(\sqrt{A/\pi}\)

Key Concepts

Understanding Circle Area

The concept of circle area is important in geometry and real-life applications alike. The area of a circle tells you the amount of space enclosed within its boundary. Calculating the area of a circle involves a simple formula:

• The formula is given by: \[ A = \pi r^2 \]where \(A\) represents the area and \(r\) is the radius of the circle.
• \(\pi\) (pi) is a constant, approximately 3.14159. It is a unique mathematical constant that represents the ratio of the circumference of any circle to its diameter.
• Understanding this formula is crucial as it forms the basis for solving more complex geometric problems.

By inserting the radius into the formula, you can calculate the area. Conversely, if you know the area and want to find the radius, as in this problem, you need to reorganize the equation. This is where isolating variables comes into play.

Isolating Variables

When solving equations, isolating the variable you need is a key step. Isolating variables means manipulating the equation so that this variable is alone on one side of the equation. This is particularly useful when you need to solve for a specific quantity, like finding the circle’s radius when its area is given.

• To begin with, identify the variable you need to solve for. In this exercise, it's the radius \(r\).
• Next, apply inverse operations to both sides of the equation to free the variable. For instance, if it’s multiplied by something, you would divide both sides by that same thing.
• In our circle area problem, divide both sides by \(\pi\) to isolate \(r^2\), thereby obtaining:\[ \frac{A}{\pi} = r^2 \]

After isolating \(r^2\), the next step is to solve for \(r\). This is achieved through the application of a square root.

Applying the Square Root

The square root is a fundamental concept in mathematics, often used to simplify or solve equations involving squares. In this context, applying the square root helps us find the radius \(r\) from the equation \(r^2\).

• The principle behind taking the square root of both sides of an equation is to reverse the squaring process. This isolates \(r\), giving us:\[ r = \sqrt{\frac{A}{\pi}} \]
• While solving, only consider the principal square root, which is the positive root, as instructed by the problem's conditions.
• Understanding this operation is vital, as taking the square root can also introduce extraneous roots in other problems where negative values are valid.

This entire process of division and square root application is crucial in problems requiring solving for a squared variable, reaffirming the concept of non-negative quantities in real-world applications.