Problem 57 Solve each formula for the indic... [FREE SOLUTION]

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Chapter 3: Problem 57

Solve each formula for the indicated variable. $$ $n=\frac{c}{v}$ for $c$ (physics)$$

Short Answer

Expert verified

The given formula is $n = \frac{c}{v}$. To solve for $c$, multiply both sides by $v$ to get $c = vn$.

Step by step solution

Write down the given formula.

The given formula is: $n = \frac{c}{v}$

Multiply both sides by $v$ to eliminate the denominator.

We want to get $c$ by itself. Since $c$ is currently in the denominator, we can multiply both sides of the equation by $v$ to eliminate the denominator: $v \cdot n = v \cdot \frac{c}{v}$

Simplify the equation.

After multiplying both sides by $v$, we get: $vn = c$

Write down the final solution.

The given formula, solved for the variable $c$, is: $c = vn$

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Key Concepts

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

Variable Isolation

Variable isolation is an essential part of solving equations, especially when you want to find the value of a particular variable. Let's break down what we mean by isolating a variable.First, understand that isolating a variable means getting the variable by itself on one side of the equation. This often involves removing any other numbers or variables that are attached to it.

Identify the target variable you need to isolate. In our example, it is the variable $c$.
Determine what operations are holding the variable where it is. For instance, in $n = \frac{c}{v}$, $c$ is divided by $v$.
Perform inverse operations to move those parts to the other side of the equation, so only the variable remains. Here, multiplying both sides by $v$ cancels the division and frees $c$.

Remember, the operations you choose to isolate the variable should balance the equation鈥攚hatever you do to one side, you must do to the other. This keeps the equation true.

Physics Formulas

Physics formulas often involve variables that represent physical quantities, like speed, time, or in this case, the refractive index $n$, speed of light $c$, and velocity $v$.Understanding each variable's role is crucial:

$n$ denotes the refractive index. It reflects how much the speed of light is reduced inside a medium compared to a vacuum.
$c$ represents the speed of light in the medium, and it's our target variable in this example.
$v$ is the velocity of light in the medium.

Physics formulas like $n = \frac{c}{v}$ can be rearranged depending on what information you need. Think of them as tools鈥攜ou might need to find the speed of light in different conditions, or perhaps how light interacts with different materials. Solving these formulas requires a solid grasp of algebraic manipulation, similar to the steps we outlined.

Rational Equations

Rational equations involve fractions containing one or more variables in the numerator, denominator, or both.A critical step in solving rational equations is to eliminate the fractions鈥攐ften the trickiest part. Here's how it works:

Identify the fractions in the equation. In our example, the equation begins as $n = \frac{c}{v}$.
Eliminate the fraction by finding a common denominator or, as frequently done, multiplying through by the current denominator to clear the fractions.
Apply algebraic operations to solve for the desired variable. In this case, multiplying both sides by $v$ not only eliminates the fraction but simplifies our expression to isolate $c$.

The goal is to transform the rational equation into a simpler one, with no fractions, to make the solving process more straightforward. Each step must maintain the equation鈥檚 balance鈥攅very action performed on one side must be mirrored on the other. This ensures that the solution remains accurate.

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91影视

Solve each formula for the indicated variable. $$ \(n=\frac{c}{v}\) for \(c\) (physics)$$

Short Answer

Step by step solution

Write down the given formula.

Multiply both sides by \(v\) to eliminate the denominator.

Simplify the equation.

Write down the final solution.

Key Concepts

Variable Isolation

Physics Formulas

Rational Equations

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