Problem 67 - Algebra for College Students

Question

For the indicated variable. Assume that all letters represent positive numbers. \(A=\pi r^{2}\) for \(r\)

Step-by-Step Solution

Answer

The solution for \(r\) is \(r = \sqrt{\frac{A}{\pi}}\).

Step 1: Understand the Formula

The given formula is the area of a circle: \(A = \pi r^2\). Here, \(A\) represents the area, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius of the circle. Our goal is to solve for \(r\).

Step 2: Isolate \(r^2\)

To solve for \(r\), first isolate \(r^2\) on one side of the equation. Divide both sides of the equation by \(\pi\):\[\frac{A}{\pi} = r^2\]

Step 3: Solve for \(r\)

To find \(r\), take the square root of both sides of the equation. Remember to consider only the positive square root since \(r\) represents a radius, which is always positive:\[r = \sqrt{\frac{A}{\pi}}\]

Step 4: Check Your Work

Verify by substituting the expression for \(r\) back into the original equation: \(\pi r^2\). If you get back \(A\), then the solution is correct. After substitution, we have:\[\pi \left(\sqrt{\frac{A}{\pi}}\right)^2 = A\]Simplifying the left side gives \(A\), confirming the solution is correct.

Key Concepts

Understanding the Area of a Circle

The area of a circle is a fundamental concept in geometry. It helps us determine how much space is contained within a circular boundary. The formula used to find this area is \(A = \pi r^2\). Here:

• \(A\) stands for the area of the circle.
• \(\pi\) (pi) is a mathematical constant, approximately equal to 3.14159. It represents the ratio of the circumference of any circle to its diameter.
• \(r\) is the radius of the circle, which is the distance from the center of the circle to any point on its edge.

To find the area, you square the radius (multiply it by itself) and then multiply by \(\pi\). This gives you the total space within the circle's boundary. Remember, more space means a larger area. A larger radius translates into a larger area!
Being familiar with this formula allows you to calculate the area easily whenever you know the radius, and it also becomes the basis for more complex geometrical calculations.

Isolating Variables in Equations

In algebra, isolating a variable means rearranging an equation so that the variable stands alone on one side. This process is crucial when solving for unknowns. Let's see how this applies to our equation for the area of a circle.
We have:\[A = \pi r^2\]
and we want to solve for \(r\). Here's how we isolate \(r^2\):

• Step 1: Divide both sides by \(\pi\) to get \(r^2\) alone:\[\frac{A}{\pi} = r^2\]

By dividing by \(\pi\), you remove the constant from the side of the equation with \(r^2\). This means you are simplifying the equation down to the clear relationship between \(A\) and \(r^2\). Isolating variables is a critical skill as it transforms complicated equations into something more approachable, revealing the core relationship between the terms involved.

Taking the Square Root

Finding a square root is a common operation in algebra, especially when you are solving equations involving squares. When you have an equation like \(r^2 = \frac{A}{\pi}\), the next step is to find \(r\) by "undoing" the square with a square root.
To find \(r\), you take the square root of both sides:\[r = \sqrt{\frac{A}{\pi}}\]

• The square root symbol (\(\sqrt{}\)) represents a number that gives the original value when multiplied by itself.
• In the context of this problem, since \(r\) is a radius, we only consider the positive root.

Imagine solving a simple puzzle – taking the square root gives you one of the original numbers you "squared" to get that result. This concept is not only key in algebra but is also used in geometry, calculus, and many areas of science.
Understanding how to correctly apply square roots in equations is important for solving not just mathematical problems but also real-world issues where measurement errors need to be corrected.