Problem 12 - Algebra A Combined Function

Question

Represent each given condition using a single variable, \(x\). The area of the circle is \(25 \pi\) square kilometers. Find its radius.

Step-by-Step Solution

Answer

The radius of the circle is 5 kilometers.

Step 1: Identify the Formula for the Circle's Area

The area of a circle is given by the formula: \( A = \pi r^2 \), where \( r \) is the radius.

Step 2: Set the Formula Equal to the Given Area

Given that the area is \( 25 \pi \), set the formula equal to the given area: \( \pi r^2 = 25 \pi \).

Step 3: Simplify the Equation

Divide both sides of the equation by \( \pi \) to eliminate it from the equation, resulting in \( r^2 = 25 \).

Step 4: Solve for the Radius

To find \( r \), take the square root of both sides of the equation: \( r = \sqrt{25} \).

Step 5: Calculate the Value of the Radius

Simplify \( r = \sqrt{25} \) to find \( r = 5 \).

Key Concepts

Circle Area Formula

When dealing with circles, one of the most fundamental ideas is how to calculate the area. The formula for this is \[ A = \pi r^2 \] where \( A \) represents the area, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.
This simple equation allows us to calculate how much space is contained within the boundary of a circle. You can think of the radius \( r \) as the distance from the center of the circle to any point along its edge. Understanding this formula is key to solving many problems involving circles.

• Reminder: Always ensure you understand what each symbol in a formula represents.
• Know that \( \pi \) is a universal constant in circle calculations.

By memorizing and understanding the circle area formula, you can tackle numerous geometry problems effectively.

Solving for Radius

Solving for the radius of a circle is often necessary when you know the area. To find the radius from the circle's area, the circle area formula is rearranged. Given the formula \( A = \pi r^2 \), you can solve for \( r \) when \( A \) is known. This process involves setting the area you know equal to the formula and isolating \( r \).

• For example, start with the formula: \( \pi r^2 = 25\pi \).
• Then, divide both sides by \( \pi \) to simplify: \( r^2 = 25 \).
• Next, take the square root of both sides to isolate \( r \): \( r = \sqrt{25} \).

This yields \( r = 5 \), meaning the radius of the circle is 5 units. Solving for the radius by rearranging and simplifying equations helps in understanding the key relationship between the radius and the circle's area.

Simplifying Equations

Simplifying equations is a useful skill in mathematics that helps make equations easier to work with and solve. It involves manipulating the equation in ways that make it less complex without changing its fundamental meaning.
In our scenario, you start by having an equation with the area of the circle set equal to the formula: \( \pi r^2 = 25 \pi \). To simplify, remove common factors from both sides. Since \( \pi \) appears on both sides of the equation, you can divide through by \( \pi \) to simplify it to \( r^2 = 25 \). This eliminates the \( \pi \), leaving us with a much simpler equation to solve for \( r \).

• Look for common terms on both sides of the equation when simplifying.
• Remember that simplifying equations makes it easier and faster to find solutions.

Simplifying not only streamlines processes but also aids in appreciating the elegance and efficiency of mathematical procedures.