Problem 118 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s l=\pi r^{2} \text { for } r$$

Step-by-Step Solution

Answer

\(r = \pm \sqrt{\frac{sl}{\pi}}\).

Step 1: Isolate the term containing the variable

We start with the equation \(sl = \pi r^2\). Our goal is to solve for \(r\). To isolate the term that contains \(r\), divide both sides by \(\pi\). This gives \(r^2 = \frac{sl}{\pi}\).

Step 2: Solve for the variable by taking the square root

Now that we have \(r^2 = \frac{sl}{\pi}\), we can solve for \(r\) by taking the square root of both sides. This results in \(r = \pm \sqrt{\frac{sl}{\pi}}\). The \(\pm\) sign indicates that there are two possible values for \(r\): one positive and one negative.

Key Concepts

Formula Rearrangement

In algebra, formula rearrangement is a fundamental skill that involves manipulating equations to solve for a specific variable. To rearrange a formula, one must understand the desired target and systematically perform operations that isolate that variable. In our problem, the goal was to solve for the variable \( r \) in the equation \( sl = \pi r^2 \).

Here’s how we approach formula rearrangement:

• Identify the term that contains the variable you wish to isolate, which is \( r^2 \).
• Perform an operation to both sides of the equation to isolate this term. In this case, we divide both sides by \( \pi \) to obtain \( r^2 = \frac{sl}{\pi} \).

By following these steps, we transform the equation closer to making \( r \) the subject. Proper formula rearrangement sets the stage for further operations, like solving for \( r \) by introducing square roots.

Square Roots

Taking square roots is a key technique in algebra, particularly when solving quadratic terms. In the exercise, once we reach \( r^2 = \frac{sl}{\pi} \), it becomes crucial to apply the square root to both sides of the equation to express \( r \).

Here’s what you need to keep in mind:

• Applying the square root to both sides gives \( r = \pm \sqrt{\frac{sl}{\pi}} \). The \( \pm \) symbol indicates two possible solutions for \( r \): one positive and one negative.
• Taking the square root is an essential step when handling equations where the variable is squared, as is often the case in circular areas and volumes.

Students should understand that square roots can yield dual results in real number solutions, which are vital in contexts such as physics, engineering, and geometry. This dual nature is why \( \pm \) is important and must be shown unless context dictates otherwise.

Variable Isolation

Variable isolation is the ultimate goal of the algebraic manipulation process when solving equations. After rearranging the given formula and applying square roots, the goal is to have \( r \) alone on one side of the equation.

Why is this important?

• It simplifies the equation, making \( r \) explicitly defined in terms of other known quantities \( sl \) and \( \pi \).
• An isolated variable allows us to directly compute its value from given data without ambiguity.

For instance, once \( r \) is isolated in \( r = \pm \sqrt{\frac{sl}{\pi}} \), this means the radius of a circle can be calculated immediately if \( s \) and \( l \) are known. It highlights how variable isolation is crucial for practical problem-solving and making predictions or inferences based on mathematical models.