Problem 13 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(V=l w h\) for \(w\)

Step-by-Step Solution

Answer

\(w = \frac{V}{l h}\)

Step 1: Identify the Formula

The formula given is for the volume of a rectangular prism: \(V=l w h\). We need to solve for \(w\).

Step 2: Isolate the Variable

To isolate \(w\), we need to divide both sides of the equation by the terms that are multiplied with \(w\), which are \(l\) and \(h\).

Step 3: Perform the Division

Divide both sides of the equation \(V=l w h\) by \(l h\) to get \(w\) alone on one side. This gives us \(w=\frac{V}{l h}\).

Step 4: Solution Simplification

Verify that the formula \(w=\frac{V}{l h}\) correctly isolates \(w\) and represents the width in terms of volume, length, and height.

Key Concepts

Solving for a Variable

When working with algebraic formulas, such as those used to calculate volume, we often need to solve for a particular variable. This means rearranging the equation so that the desired variable is isolated on one side. The process involves using basic algebraic operations like addition, subtraction, multiplication, and division.

In the example given, we have the volume formula for a rectangular prism, which is: \[ V = lwh \] Here, the goal is to solve for the variable \(w\), representing the width of the prism. The technique used is division, which helps us "break down" the multiplication, this means we have to divide both sides of the equation by the other factors, which are \( l \) and \( h \). Doing so gives: \[ w = \frac{V}{lh} \] This new equation perfectly isolates \(w\), showing us how wide the prism is when we know its volume, length, and height.

Volume Formula

The formula for the volume of a rectangular prism is a fundamental concept in geometry and algebra. It tells us the amount of space that fits inside a three-dimensional shape. When we say volume, we mean the number of cubic units that fully occupy the interior of the shape.

The volume formula for a rectangular prism is given by: \[ V = lwh \] where:

• \(V\) is the volume,
• \(l\) is the length,
• \(w\) is the width,
• \(h\) is the height.

In practical terms, this formula helps us calculate how much space an object occupies. It's particularly useful in fields like engineering, construction, and packaging, where understanding the capacity of a space is critical.

Remember, all dimensions must be in the same units to ensure the volume is accurate. If you're measuring in feet, for instance, everything should be in feet to find the volume in cubic feet.

Rectangular Prism

A rectangular prism is a three-dimensional solid shape that has six faces, all of which are rectangles. It's one of the most common and easiest shapes to understand in solid geometry.

Each rectangular prism is defined by three dimensions:

• Length ( l) : The longest side of the rectangle.
• Width ( w) : The side of the rectangle that's perpendicular to the length.
• Height ( h) : The dimension that goes upwards, perpendicular to both the length and width.

In real life, rectangular prisms are everywhere. Think of a brick, a shoebox, or a book; all these are examples of rectangular prisms. By knowing these three dimensions, you can calculate not only the volume but other properties like surface area.

A helpful tip is to visualize a rectangular prism as stacking sheets of rectangular paper up as high as the height measurement. This visualization helps in understanding both the structure and the calculation of volume through simple multiplication of its dimensions.