Problem 71 - Elementary Algebra

Question

The volume of a rectangular solid is given by \(V=l w h,\) where \(/\) represents the length, w represents the width, and \(h\) is the height of the solid. Find the volume of a rectangular solid if the length is 2 inches, the width is 3 inches, and the height is 4 inches.

Step-by-Step Solution

Answer

The volume of the rectangular solid is 24 cubic inches.

Step 1: Identify the given values

First, note down the values provided in the problem. The length of the rectangular solid is given as \( l = 2 \) inches, the width is \( w = 3 \) inches, and the height is \( h = 4 \) inches.

Step 2: Recall the formula for volume

The volume \( V \) of a rectangular solid is calculated using the formula \( V = l w h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Step 3: Substitute the given values into the formula

Replace the variables in the formula with the given values: \( l = 2 \), \( w = 3 \), and \( h = 4 \). Thus, \( V = 2 \cdot 3 \cdot 4 \).

Step 4: Calculate the volume

Perform the multiplication to find the volume. Start with multiplying the length and width: \( 2 \cdot 3 = 6 \). Then multiply the result by the height: \( 6 \cdot 4 = 24 \). Thus, the volume of the rectangular solid is 24 cubic inches.

Key Concepts

rectangular solid

A rectangular solid, also known as a rectangular prism, is a three-dimensional shape that has six faces. All of these faces are rectangles. This type of solid has several characteristic properties:

• There are three sets of parallel faces, and each set consists of opposite rectangles that are equal in size.
• The face-to-face intersection lines are known as the edges.
• Another important feature is its vertices, which are the corner points where edges meet.

The shape is defined by three primary measurements: length, width, and height. These dimensions help determine its size and volume. In real life, objects like boxes and rooms closely resemble rectangular solids. Understanding their structure aids in solving volume-related problems.

volume formula

The volume formula for a rectangular solid is essential for calculating how much space or capacity the shape encloses. The formula is given by \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

This formula can be visualized by considering how multiple layers of rectangular areas "stack" up to fill the space within the solid. Here's a breakdown:

• The term \( l \times w \) calculates the area of the base, the bottom face of the rectangular solid.
• When you multiply this base area by \( h \), the height, you're effectively summing up all these base layers running through the height.

In the exercise, by substituting \( l = 2 \), \( w = 3 \), and \( h = 4 \), we calculate the volume as 24 cubic inches. This means the solid can contain or occupy 24 cubes of 1 cubic inch each.

mathematical problem-solving

Mathematical problem-solving involves strategically breaking down a problem into smaller steps to find a solution efficiently. This process is particularly useful when dealing with geometric calculations, such as finding the volume of a rectangular solid. Here’s a useful approach:

1. **Understand and Identify**: Clearly state the problem by identifying all given values. In this exercise, recognize \( l = 2 \), \( w = 3 \), and \( h = 4 \).

2. **Apply the Formula**: Use relevant mathematical formulas, like \( V = l \times w \times h \). By familiarizing oneself with these, solving similar problems becomes routine.

3. **Substitute and Calculate**: Carefully substitute known values into the equation and follow through with the arithmetic operations. Multiply in steps—first \( 2 \times 3 \) to get 6, then \( 6 \times 4 \) to determine the total volume.

4. **Review**: Always revisit each step to ensure the calculations are correct and make logical sense. This approach helps in preventing small errors that could lead to incorrect answers.

By consistently practicing these methods, solving volume problems and other mathematical challenges becomes easier and more intuitive.