Problem 63 - Precalculus: Functions and Graphs

Question

Solve for the specified variable. $$F=\frac{\pi P R^{4}}{8 V L} \text { for } R$$

Step-by-Step Solution

Answer

\( R = \sqrt[4]{\frac{8 F V L}{\pi P}} \)

Step 1: Understand the Equation

The equation given is: \[ F = \frac{\pi P R^{4}}{8 V L} \]We need to solve this equation for the variable \( R \). This means isolating \( R \) on one side of the equation.

Step 2: Multiply Both Sides by the Denominator

To remove the fraction, multiply both sides by \( 8 V L \):\[ F \times 8 V L = \pi P R^{4} \]This simplifies to:\[ 8 F V L = \pi P R^{4} \]

Step 3: Divide by the Coefficient of \( R^4 \)

Now divide both sides by \( \pi P \) to isolate \( R^4 \):\[ \frac{8 F V L}{\pi P} = R^{4} \]

Step 4: Take the Fourth Root

To solve for \( R \), take the fourth root of both sides:\[ R = \sqrt[4]{\frac{8 F V L}{\pi P}} \]This gives us \( R \) in terms of the other variables.

Key Concepts

Equation Manipulation

Equation manipulation is a crucial skill in algebra, as it allows you to rearrange an equation to solve for a particular variable. In our exercise, manipulating the equation involves dealing with a fraction and rearranging terms to isolate the desired variable.

Start by identifying what you need to solve for—in this case, it's the variable \( R \). The original equation is in the form \( F = \frac{\pi P R^{4}}{8 V L} \). Notice that \( R \) is part of a larger fraction. To solve for \( R \), you must first "undo" this fraction to free \( R \) from the denominator.

Multiply both sides of the equation by the entire denominator \( 8 V L \). This step is critical in removing the fraction:

• \( F \times 8 V L = \pi P R^{4} \)

By transposing the fraction out of the equation, it becomes much easier to continue solving for \( R \). Ensure that each step is balanced by performing operations equally on both sides.

Algebraic Isolation

Algebraic isolation refers to the process of getting the variable you’re solving for on its own on one side of the equation. When isolating a variable, you undo any addition, subtraction, multiplication, or division that’s affecting that variable.

In our example, after removing the fraction, the equation simplifies to \( 8 F V L = \pi P R^{4} \). Now, the goal is to have \( R^{4} \) by itself. To accomplish this, divide both sides by \( \pi P \), which is the coefficient accompanying \( R^{4} \):

• \( \frac{8 F V L}{\pi P} = R^{4} \)

This step isolates \( R^{4} \) on one side of the equation. When isolating a variable, it’s important to work systematically, performing inverse operations until the variable is fully isolated. This ensures that each step you take moves you closer to having the variable alone.

Exponentiation and Roots

Exponentiation and its inverse operation, roots, are used to manage powers when solving equations. An exponent indicates how many times a number is multiplied by itself. Conversely, taking a root undoes this multiplication, which is essential when a variable is raised to a power.

In this problem, \( R \) is raised to the fourth power (\( R^{4} \)). To "undo" this, you need to take the fourth root of both sides. This operation will solve for \( R \):

• \( R = \sqrt[4]{\frac{8 F V L}{\pi P}} \)

Taking roots balances the operation, reducing the power of the variable, and thereby isolating it. This procedure is used frequently in algebra when dealing with powers; knowing how to apply it properly is crucial for solving equations with variables raised to any power.