Problem 35 - Prealgebra

Question

As you know, the volume \(V\) enclosed by a rectangular solid with length \(I,\) width \(w,\) and height \(h\) is \(V=I \cdot w \cdot h .\) Find \(V\) if: \(I=6\) inches, \(w=12\) inches, and \(h=5\) inches

Step-by-Step Solution

Answer

The volume is 360 cubic inches.

Step 1: Understand the Formula

The volume of a rectangular solid is given by the formula \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height. We need to plug in the given values into this formula.

Step 2: Substitute Given Values

Insert the given values into the formula: \( l = 6 \) inches, \( w = 12 \) inches, and \( h = 5 \) inches. Thus, the equation becomes \( V = 6 \times 12 \times 5 \).

Step 3: Multiply Length by Width

First, multiply the length by the width: \( 6 \times 12 = 72 \).

Step 4: Multiply Result by Height

Now, take the result from the previous step and multiply it by the height: \( 72 \times 5 = 360 \).

Step 5: State the Volume

Therefore, the volume of the rectangular solid is 360 cubic inches.

Key Concepts

Volume Calculation

Understanding how to calculate the volume of a rectangular solid is both fun and practical! The volume represents the amount of space that a shape occupies and is usually measured in cubic units like cubic inches or cubic centimeters.
To calculate the volume, we use the formula: \[ V = l \times w \times h \]
Where:

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

The key to remember is that these measurements must be in the same unit, such as inches or centimeters. Once you have the proper formula and measurements, it's a simple matter of multiplication to find the volume of any rectangular solid.

Rectangular Prism

A rectangular prism, also known as a rectangular solid, is a 3D shape that is made up of six rectangular faces. It's somewhat like a box.
This specific shape is characterized by having:

• Opposite faces that are equal in size,
• All internal angles are right angles (90 degrees),
• Three pairs of parallel faces (like a shoe box).

This solid shape is common in our everyday lives, from shoe boxes and books to buildings and furniture.
When working with a rectangular prism, identifying the length, width, and height is crucial for performing volume calculations.
This basic geometric understanding is essential as it allows anyone to calculate how much space the object occupies.

Multiplication of Dimensions

The multiplication of dimensions is the heart of finding the volume of a rectangular prism. This involves multiplying three important measurements: length, width, and height.
Here's how this process works step-by-step:
First, multiply the length by the width. For example, if the length is 6 inches and the width is 12 inches, you would get \[ 6 \times 12 = 72 \] square inches.
Next, take this result and multiply it by the height. If the height is 5 inches, then: \[ 72 \times 5 = 360 \] cubic inches.
Each measurement must be multiplied consecutively to ensure you are calculating the volume correctly.
These steps highlight a simple rule: multiplying these dimensions gives the total space inside the rectangular solid, perfectly illustrating why careful calculation is crucial.