Problem 62 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(V=L W H\) for \(L \quad\) (Volume of a box)

Step-by-Step Solution

Answer

The formula solved for \( L \) is \( L = \frac{V}{W \cdot H} \).

Step 1: Understand the formula

The formula provided represents the volume of a rectangular box, where \( V \) is the Volume, \( L \) is the Length, \( W \) is the Width, and \( H \) is the Height. Our task is to solve this equation for \( L \).

Step 2: Isolate the variable

To solve for \( L \), we need to isolate it on one side of the equation. The current equation is \( V = L \cdot W \cdot H \). In order to isolate \( L \), divide both sides of the equation by \( W \cdot H \).

Step 3: Rewrite the formula

After dividing both sides by \( W \cdot H \), the equation becomes \( \frac{V}{W \cdot H} = L \). This expresses \( L \) in terms of \( V \), \( W \), and \( H \).

Key Concepts

Volume of a Box

When we talk about the volume of a box, we're essentially discussing the amount of space inside it. Imagine filling a box with water or packing it with little cubes of a standardized size. The measure of all that water or all those small cubes is what we call volume.

In mathematical terms, the volume of a rectangular box or rectangular prism can be calculated using the formula:

- \( V = L \cdot W \cdot H \)

Here,

• \( V \) represents the volume,
• \( L \) is the length,
• \( W \) is the width, and
• \( H \) is the height of the box.

This formula gives a quantitative measure of the space within the box based on its dimensions. Understanding this basic formula is crucial for solving more complex problems involving volume in geometry.

Solving for a Variable

Mathematics often requires us to solve for a particular variable. Solving for a variable means adjusting an equation so that you have that variable alone on one side of the equation. This process helps in understanding how changes in other variables affect the variable of interest.

In our example, we started with the formula for the volume of a box: \( V = L \cdot W \cdot H \). Our task was to solve for \( L \), meaning we wanted to express \( L \) in terms of the other variables \( V \), \( W \), and \( H \).

Often, solving for a variable involves these steps:

• Identify what variable you need to solve for.
• Use algebraic manipulation to get that variable by itself. This could include using addition, subtraction, multiplication, or division.

These principles are useful not just in geometry, but in various fields such as physics, engineering, and economics.

Isolating a Variable

Isolating a variable is a vital skill in algebra. It involves performing operations to rewrite an equation so that one specific variable stands alone on one side.

In the context of our volume of a box equation, isolating \( L \) means rewriting the equation to express \( L \) in terms of the other parameters. Here are the steps you would take:

• Start with the original equation: \( V = L \cdot W \cdot H \).
• To isolate \( L \), divide each side of the equation by \( W \cdot H \).
• This results in \( L = \frac{V}{W \cdot H} \), successfully isolating \( L \).

This allows you to understand how \( L \) changes with variations in volume or dimensions of width and height. Mastering this technique is critical, as it forms the backbone of solving many mathematical and real-world problems where you need a clear understanding of relationships between different quantities.