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Chapter 3: Problem 58

Solve each formula for the indicated variable. $$f=\frac{R}{2}\( for \)R$ (physics)$$

Short Answer

Expert verified
The short answer to solving the formula \(f=\frac{R}{2}\) for \(R\) is: \(R = 2f\).

Step by step solution

01

Identify the formula given

The given formula is \(f=\frac{R}{2}\), where \(f\) and \(R\) are the variables.
02

Rearrange the equation to isolate \(R\)

To isolate \(R\), we need to multiply both sides of the equation by 2. This will eliminate the denominator 2 from the right side of the equation. $$2 \cdot f = \frac{R}{2} \cdot 2$$
03

Simplify the equation

After multiplying both sides of the equation by 2, we get: $$ 2f = R$$
04

Final Expression for \(R\)

We have successfully isolated R on one side of the equation. The final expression for \(R\) in terms of \(f\) is: $$ R = 2f$$

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. Equations come in various forms, but the goal is always to isolate the variable. Here are a few key steps:
  • Understand the equation: Determine what is given and what is to be found.
  • Perform inverse operations: Use addition, subtraction, multiplication, or division to isolate the variable.
  • Check the solution: Substitute back into the original equation to verify the answer.
In our original exercise, the equation was solved by multiplying both sides by 2 to eliminate the fraction. This step makes it straightforward to identify what needs to be undone to isolate the desired variable.
Formula Rearrangement
Rearranging formulas is modifying how variables are presented or expressed without changing their relationships. It's especially useful when you want to express a variable in terms of others.
Consider the following techniques:
  • Identify the target variable you want to isolate.
  • Use algebraic operations to systematically move terms around, always doing the same operation to both sides of the equation.
  • Simplify the equation to make the target variable the subject.
In our solution example, rearranging involved multiplying both sides of the equation by 2, which moved from a fraction into a simpler expression with the variable R on one side alone.
Physics Formulas
Physics formulas are tools that express relationships between different physical quantities. Students often come across them in studies involving forces, motion, or energy. Learning how to rearrange and solve these formulas is crucial for understanding physics problems.
Here are some tips when working with physics formulas:
  • Understand the physical meaning of each variable. This will help make sense of why you need to rearrange the formula.
  • Ensure dimensional consistency: The units should match in the rearranged formula. This is a quick check for accuracy.
  • Recognize patterns: Many physics problems use similar foundational concepts, so recognizing patterns helps in rearranging the formulas correctly.
The formula in our exercise relates the force (\( f \)) to another physical quantity (\( R \)). Rearranging it correctly reveals what each variable depends on, emphasizing the importance of understanding these relationships in physics.

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