Problem 63 - Elementary and Intermediate Algebra

Question

Solve each equation for the specified variable or expression. $$ v=\sqrt{2 g h} \text { for } h $$

Step-by-Step Solution

Answer

\( h = \frac{v^2}{2g} \)

Step 1: Isolate the Squared Term

To solve for \( h \), start by getting rid of the square root. Square both sides of the equation: \[ v^2 = (\sqrt{2gh})^2 \] This simplifies to: \[ v^2 = 2gh \]

Step 2: Solve for h

Now that we have \( v^2 = 2gh \), we need to isolate \( h \). Divide both sides of the equation by \( 2g \): \[ h = \frac{v^2}{2g} \] This gives us the value of \( h \) in terms of \( v \) and \( g \).

Key Concepts

Variables

Variables are letters or symbols that represent an unknown number or value in mathematics. They are placeholders that allow us to describe general relationships between quantities. For example, in the equation \( v = \sqrt{2gh} \), the letters \( v \), \( g \), and \( h \) are variables.
Each variable can take different values depending on the situation or context. In our original equation, the goal is to find the value of one specific variable, \( h \), in terms of other variables.
To manage variables effectively, follow these general tips:

• Identify the variables in an equation.
• Understand which variable you need to solve for.
• Treat variables as numbers when performing operations like addition, subtraction, multiplication, and division.
• Manipulate the equation using algebraic operations to express the desired variable in terms of others.

This understanding is essential for problem-solving in algebra since it allows for the formulation of equations that model real-world situations.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol \( \sqrt{\cdot}\).
In equation solving, encountering square roots typically requires squaring both sides to eliminate the root. Squaring each side of an equation allows us to work with non-square terms, simplifying complex equations.
Consider the equation \( v = \sqrt{2gh} \). To eliminate the square root:

• Square both sides to remove the square root, resulting in \( v^2 = 2gh \).
• Squaring is a reversal process that simplifies equations for easier manipulation.
• Always remember, when squaring, any solutions should be checked in the original equation, as squaring can sometimes introduce extraneous solutions.

Understanding square roots is crucial for manipulating equations effectively, especially when isolating variables.

Isolating Terms

Isolating terms in an equation means moving a specific term to one side of the equation to solve for it. By isolating a term, you can focus on finding its value more effectively.
To isolate, you need to perform inverse operations that "undo" operations in the equation. For example, multiplication can be undone with division.
In the equation \( v^2 = 2gh \), to solve for \( h \):

• Divide both sides by \( 2g \) to isolate \( h \), leading to \( h = \frac{v^2}{2g} \).
• This involves performing the opposite operation (division, in this case) to isolate \( h \).

When isolating terms, carefully track each operation—such as multiplication or division—used to manipulate the equation and arrive at a solution. By mastering this process, you can solve for any variable within a given equation, simplifying complex mathematical problems.