Problem 12 - Algebra 1

Question

Solve for the indicated variable. Volume of a Rectangular Prism Solve for \(h: V=\ell w h\)

Step-by-Step Solution

Answer

The isolated variable \(h\) in the volume formula for a rectangular prism is \(h = \frac{V}{\ell w}\)

Step 1: Understand the Formula

Firstly, the formula for the volume of a rectangular prism; \(V = \ellwh\), needs to be understood. In the formula, \(V\) stands for Volume, \(\ell\) for Length, \(w\) for Width, and \(h\) for Height. We are required to isolate \(h\) in this equation.

Step 2: Divide Both Sides by \(\ell w\)

To isolate \(h\), we must eliminate other variables from the left side, where \(h\) is currently located. We can achieve this by dividing both sides of the equation by \(\ell w\). This operation will give \(h = \frac{V}{\ell w}\).

Step 3: Writing Down the Final Solution

After successfully isolating \(h\) in the equation, the final step is to record the new expression for \(h\), which is \(h = \frac{V}{\ell w}\). This is the expression for height in terms of volume, length, and width.

Key Concepts

Solving for a Variable

Solving for a variable involves rearranging an equation so that the desired variable is isolated on one side of the equation. In the given exercise, we aim to solve for height, represented by the letter \( h \), in the volume formula of a rectangular prism.

• We start with the equation \( V = \ell w h \) where \( V \) is volume, \( \ell \) is length, \( w \) is width, and \( h \) is height.
• To solve for \( h \), it means we want to make \( h \) the subject of the equation.

By doing this, you learn how to isolate a variable, a skill that is critical in algebra and helps you understand the relationship between the variables involved in a formula.

Formula Manipulation

Formula manipulation is a skill that allows us to rearrange equations to express a quantity in terms of others. Let's take a closer look at how to manipulate the volume formula of a rectangular prism to solve for \( h \): Once you understand the initial formula \( V = \ell w h \), you can transform it to isolate \( h \).

• The goal is to have \( h \) by itself on one side. To achieve this, divide both sides by the product \( \ell w \) which currently multiplies \( h \).
• This operation gives us: \( h = \frac{V}{\ell w} \).

The ability to manipulate formulas is essential in solving equations and adapting them to solve real-world problems. Familiarity with formula manipulation can make it easier to tackle complex mathematical problems by breaking them down into simpler steps.

Geometry

Geometry is a branch of mathematics that deals with shapes and the properties of space. In this context, we're exploring a rectangular prism, which is a three-dimensional shape defined by its length, width, and height. Understanding the geometry of a rectangular prism helps one relate its dimensions with its volume. The formula \( V = \ell w h \) encapsulates how the length, width, and height work together to determine the space inside the prism.

• The length (\( \ell \)), width (\( w \)), and height (\( h \)) define the sides of this geometrical shape.
• Volume (\( V \)) is a measure of the amount of space within the prism.

By understanding these relationships, you gain insight into how changes in one dimension, such as the height, affect the overall volume. This geometric perspective is crucial for visualizing and solving problems involving three-dimensional objects.