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Chapter 2: Problem 22

Solve each formula for the quantity given. $$ R=\frac{k L}{d^{2}} \text { for } L $$

Short Answer

Expert verified
The expression for \( L \) is \( L = \frac{Rd^2}{k} \).

Step by step solution

01

Understand the Formula

The given formula is \( R = \frac{kL}{d^2} \). We are tasked to solve this equation for \( L \). This means we need to express \( L \) as a function of the other variables \( R, k, \text{and } d \).
02

Isolate the Fraction

To make \( L \) the subject of the formula, we need to eliminate the fraction. Start by multiplying both sides by \( d^2 \), which gives: \( Rd^2 = kL \).
03

Solve for L

Now, to solve for \( L \), we divide both sides of the equation by \( k \). This results in: \( L = \frac{Rd^2}{k} \).
04

Solution Verification

Let's verify the solution by substituting \( L = \frac{Rd^2}{k} \) back into the original equation \( R = \frac{kL}{d^2} \). We find that \( R = \frac{k (\frac{Rd^2}{k})}{d^2} \), which simplifies back to \( R \), confirming our solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formula Solving
When solving formulas, our primary goal is to rearrange the equation to make the desired variable the subject. In this case, the task is to solve the formula \( R = \frac{kL}{d^2} \) for \( L \). To achieve this, follow these steps:
  • Begin by understanding the equation and identifying the needed variable, which is \( L \).
  • Next, manipulate the entire equation to isolate \( L \), using mathematical operations like multiplication, division, addition, or subtraction.
  • Ensure that all operations are applied to both sides of the equation to maintain balance and valid equivalence.
The outcome of formula solving ensures \( L \) is clearly expressed in terms of the other variables \( R, k, \text{and } d \). This process reveals how each component within the formula relates to one another.
Variable Isolation
Variable isolation is a critical skill in algebra that focuses on getting a specific variable alone on one side of the equation. For the formula \( R = \frac{kL}{d^2} \), our goal is to isolate \( L \). The steps to isolate \( L \) include:
  • First, eliminate the fraction by multiplying both sides by \( d^2 \). This transforms the equation to \( Rd^2 = kL \).
  • Next, to get \( L \) by itself, divide both sides by \( k \). This results in \( L = \frac{Rd^2}{k} \).
By isolating \( L \), the algebra significantly simplifies, making it easier to understand how changes in \( R, k, \, \text{and } d \) affect \( L \). This step is crucial for efficient problem-solving and easier manipulation of equations in algebra.
Equation Verification
Equation verification ensures that the solution is correct by substituting it back into the original equation. This step is essential to confirm accuracy. To verify \( L = \frac{Rd^2}{k} \) in the original formula \( R = \frac{kL}{d^2} \):
  • Substitute \( L \) in the equation. It becomes \( R = \frac{k (\frac{Rd^2}{k})}{d^2} \).
  • Simplify the expression to see whether both sides of the equation match the original formula \( R \).
After simplification, if you arrive back at \( R = R \), it confirms that the manipulation and solution are correct. Verification builds confidence and ensures the solution aligns with the original problem, reinforcing understanding and accuracy in algebraic manipulation.

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