Problem 72 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume no denominators are \(0 .\) $$\mathscr{A}=\pi r^{2}, \quad \text { for } r$$

Step-by-Step Solution

Answer

r = \sqrt{\frac{\mathscr{A}}{\pi}}

Step 1: Write down the equation

The given equation is \(\mathscr{A} = \pi r^{2}\). We are solving for the radius \(r\).

Step 2: Isolate the term with the variable

To isolate \(r^{2}\), divide both sides of the equation by \(\pi\): \(\frac{\mathscr{A}}{\pi} = r^{2}\).

Step 3: Solve for the variable

Take the square root of both sides to solve for \(r\): \(r = \sqrt{\frac{\mathscr{A}}{\pi}}\).

Key Concepts

headline of the respective core concept

Understanding the core concepts of this problem can help you solve it more easily. Let's take a deep dive into each of these concepts.

Area of a Circle

The formula for the area of a circle is \(\backslash\text{A} = \backslashpi r^{2}\). It helps us determine how much space is inside a circle.

Let's break this down:

• A: This stands for the area of the circle. Area is just a measure of the surface inside the circle's boundary.
  
• \backslashpi: This is a mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.
  
• r: The radius of the circle. The radius is the distance from the center of the circle to any point on its edge.

Remember, in geometry, knowing the relationship between a circle's area and its radius helps us in many real-world applications, like calculating the size of circular gardens or the amount of materials needed for a round table.

Isolating Variables

Isolating a variable means making it the subject of the formula or equation. It's essential for finding an unknown value.

In our initial equation, \(A = \backslashpi r^{2}\), we need to isolate \(r\). Here's the step-by-step process:

• Step 1: Identify the term with the variable you need to isolate. In this case, it's \(r^{2}\).
  
• Step 2: Perform operations that will cancel out other elements affecting the variable. For \(r^{2}\), we first divided both sides by \(\backslashpi\): \(\frac{A}{\backslashpi} = r^{2}\).[/li]
• Step 3: Keep performing these operations until the variable is by itself. In this case, we then took the square root of both sides: \(r = \backslashsqrt{\frac{A}{\backslashpi}}\).[/li]

By isolating variables, you can solve a variety of algebraic equations and determine unknown values in different contexts.

Square Root

The square root is a mathematical function that 'undoes' the effect of squaring a number. If you take the square root of \(25\), for example, you get \(5\) because \(5 \times 5 = 25\).

In algebra, the square root is symbolized as \(\backslashsqrt{ }\). A few key points to remember:

• Taking the square root of a number gives you a value that, when multiplied by itself, returns the original number.
  
• Square roots are especially useful when dealing with area calculations, as we've seen in this exercise.

In our formula for the area of a circle, solving for \(r\) required us to take the square root of both sides after isolating \(r^{2}\).

So, \(r = \backslashsqrt{\frac{A}{\backslashpi}}\) means getting the value of \(r\) by computing the square root of \(\frac{A}{\backslashpi}\).

This demonstrates the power of understanding how square roots function in algebra and geometry.