Problem 12 - Elementary Algebra

Question

Calculate the volume of a rectangular box with dimensions \(41 / 2\) feet by 6 feet by 1 foot.

Step-by-Step Solution

Answer

The volume is 123 cubic feet.

Step 1: Understand the Formula for Volume

The volume of a rectangular box (or cuboid) is found using the formula: \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Step 2: Substitute the Dimensions

Given the dimensions of the box: Length (\( l \)) is \( \frac{41}{2} \) feet, Width (\( w \)) is 6 feet, and Height (\( h \)) is 1 foot. Substitute these values into the formula: \( V = \frac{41}{2} \times 6 \times 1 \).

Step 3: Calculate the Length-Width Product

First, multiply the length with the width: \( \frac{41}{2} \times 6 = \frac{41 \times 6}{2} = \frac{246}{2} = 123 \).

Step 4: Multiply by the Height

Now, multiply the result from Step 3 by the height: \( 123 \times 1 = 123 \).

Step 5: Interpret the Result

The volume of the box is 123 cubic feet.

Key Concepts

Understanding a Rectangular Box

A rectangular box, also known as a cuboid, is a fundamental three-dimensional shape in geometry. It has six faces, all of which are rectangles. When we refer to a rectangular box, we often think of items like shoe boxes, bricks, or small pieces of furniture.
Each face of the box is a rectangle and it consists of three key dimensions:

• Length (l) - the longest side of the rectangle.
• Width (w) - the shorter side of the rectangle, perpendicular to the length.
• Height (h) - the side perpendicular to both the length and the width, which measures the extent of the box upwards.

These dimensions are essential in calculating the volume, which tells us how much space is inside the box.

Applying the Volume Formula

The volume of a rectangular box tells us how much three-dimensional space it occupies. We calculate this using the volume formula: \[ V = l \times w \times h \]Where:

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

Substituting the dimensions into this formula will help us calculate the volume. Let's consider a rectangular box with the dimensions: - Length: \( \frac{41}{2} \) feet (which is 20.5 if converted to decimal)- Width: 6 feet- Height: 1 foot
Plugging these into the formula gives us:\[ V = \frac{41}{2} \times 6 \times 1 \]This makes finding the product straightforward and leads us to solving for cubic units.

Calculating Volume in Cubic Feet

When we calculate the volume of a rectangular box using feet as the unit for length, width, and height, we express the result in cubic feet. Cubic feet represent the volume of a cube with sides that are one foot in length. This is a standard unit of measurement for volume.
By calculating the volume of the given box:1. Multiply the length by the width: \[ \frac{41}{2} \times 6 = \frac{246}{2} = 123 \]2. Multiply the result by the height: \[ 123 \times 1 = 123 \]Thus, the volume is 123 cubic feet. Cubic feet tell us how much material, substance, or objects the box can contain, or how much it can fill in space. Understanding this unit is crucial for practical applications, like knowing the storage capacity or shipping dimensions.