Problem 37 - College Algebra with Modeling and Visualization

Question

Find the volume and surface area of a rectangular box with length \(L\), width \(W\), and height \(H\). \(L=4\) feet, \(W=3\) feet, \(H=2\) feet

Step-by-Step Solution

Answer

Volume is 24 cubic feet, surface area is 52 square feet.

Step 1: Identify the Given Dimensions

We are provided with the dimensions of the rectangular box: length \(L\), width \(W\), and height \(H\). According to the problem, these are \(L = 4\) feet, \(W = 3\) feet, and \(H = 2\) feet.

Step 2: Find the Volume of the Box

The formula for the volume \(V\) of a rectangular box is \(V = L \times W \times H\). Substitute the given values to find the volume:\[V = 4 \times 3 \times 2 = 24\text{ cubic feet}\].

Step 3: Write the Formula for Surface Area

The surface area \(A\) of a rectangular box is given by the formula:\[A = 2(LW + LH + WH)\]. This formula accounts for the sum of the areas of all six faces of the rectangular box.

Step 4: Calculate the Surface Area of the Box

Using the formula, substitute the given dimensions to calculate the surface area:- Calculate \(LW\): \(4 \times 3 = 12\).- Calculate \(LH\): \(4 \times 2 = 8\).- Calculate \(WH\): \(3 \times 2 = 6\).Substitute these values into the surface area formula:\[A = 2(12 + 8 + 6) = 2(26) = 52 \text{ square feet}\].

Step 5: Summarize the Findings

The volume of the rectangular box is \(24\text{ cubic feet}\), and the surface area is \(52\text{ square feet}\).

Key Concepts

Volume Calculation

The volume of a geometric shape is essentially the amount of space it occupies. For a rectangular box, the volume is calculated using the formula \( V = L \times W \times H \). Here, \( L \) represents the length, \( W \) the width, and \( H \) the height of the box.
This is a straightforward multiplication of these three dimensions, which gives us the volume in cubic units.

• For the given box, the calculations are \( L = 4 \) feet, \( W = 3 \) feet, and \( H = 2 \) feet.
• Substituting the values, \( V = 4 \times 3 \times 2 = 24 \text{ cubic feet} \).

Understanding the concept of volume help us determine how much a box can hold, which is particularly useful in real-world applications such as packing and shipping.
Remember, the key takeaway is that volume is about space inside a box!

Surface Area Calculation

Surface area represents the total area covered by all the faces of a 3-dimensional object. For a rectangular box, it is crucial to remember that it has six faces, and the formula for calculating the total surface area is \( A = 2(LW + LH + WH) \). Here, \( LW \), \( LH \), and \( WH \) are the areas of the individual pairs of faces.

• First, compute each pair's area: \( LW = 4 \times 3 = 12 \), \( LH = 4 \times 2 = 8 \), and \( WH = 3 \times 2 = 6 \).
• Then, add them up: \( 12 + 8 + 6 = 26 \).
• Finally, multiply by 2, to account for both pairs of each face: \( A = 2 \times 26 = 52 \text{ square feet} \).

This formula effectively sums up the area of all faces, helping us understand the total surface exposure of the box, important in tasks like painting or wrapping.

Rectangular Geometry

Understanding the basic principles of rectangular geometry makes complex calculations much simpler. A rectangular box, also known as a rectangular prism, is defined by three key dimensions: length, width, and height.
These dimensions determine both the box's volume and surface area.

• Length (\( L \)): The longest side when viewed from the top.
• Width (\( W \)): The shorter side that is perpendicular to the length.
• Height (\( H \)): The vertical distance from the base to the top.

By visualizing these dimensions as the box's fundamental building blocks, you can easily apply formulas to compute both volume and surface area. Understanding rectangular geometry is essential as it lays the groundwork for more advanced mathematical studies and practical applications in everyday life, such as construction and space management.