Problem 44 - College Algebra with Modeling and Visualization

Question

Find the volume and surface area of a rectangular box with length \(L\), width \(W\), and height \(H\). \(L=8 x, W=y, H=z\)

Step-by-Step Solution

Answer

Volume is \(8xyz\). Surface area is \(16xy + 16xz + 2yz\).

Step 1: Understand the Problem

We need to find both the volume and surface area of a rectangular box. The dimensions given are: length \(L = 8x\), width \(W = y\), and height \(H = z\).

Step 2: Formula for Volume

The formula to calculate the volume \(V\) of a rectangular box is given by:\[V = L \times W \times H\]

Step 3: Substitute the Given Values into Volume Formula

Using the given dimensions, substitute into the volume formula:\[V = (8x) \times y \times z\]Simplify the expression:\[V = 8xyz\]

Step 4: Formula for Surface Area

The formula to calculate the surface area \(A\) of a rectangular box is:\[A = 2(LW + LH + WH)\]

Step 5: Substitute the Given Values into Surface Area Formula

Substitute the given dimensions into the surface area formula:\[A = 2((8x)y + (8x)z + yz)\]Simplify the expression:\[A = 2(8xy + 8xz + yz)\]\[A = 16xy + 16xz + 2yz\]

Key Concepts

Volume of Rectangular Box

Calculating the volume of a rectangular box, also known as a rectangular prism, is a straightforward process. The basic idea is to determine how much space is enclosed within the box. This is done by multiplying the three dimensions: length, width, and height. For a box with dimensions length (L), width (W), and height (H), the formula for the volume (V) is: \[ V = L \times W \times H \]For our exercise, these dimensions are given as: length = 8x, width = y, and height = z. By substituting these values into our volume formula, the calculation becomes:\[ V = (8x) \times y \times z \]When simplified, this expression results in the volume being 8xyz. In essence, multiplying these variables gives us a concise representation of how much space the box encompasses. Volume, in this context, reflects the capacity of the box to hold substances—whether liquid, solid, or gas.

Surface Area Formula

Understanding the surface area of a rectangular box is crucial when you want to know how much material is needed to cover the box, such as wrapping paper or paint. To compute the surface area, you need to consider all six faces of the box, as a rectangular box has three pairs of identical faces.The formula to determine the surface area (A) is given by:\[ A = 2(LW + LH + WH) \]This formula is derived from adding the area of each pair of faces:

• Two faces of area LW
• Two faces of area LH
• Two faces of area WH

For the given dimensions, substituting length = 8x, width = y, and height = z, the surface area calculation becomes:\[ A = 2((8x)y + (8x)z + yz) \]After simplification, this yields 16xy + 16xz + 2yz. Recognizing this expanded form helps in practical applications, ensuring all surface elements are accounted for in our computations.

Algebraic Expressions

Algebraic expressions play a crucial role in geometry, especially in calculating metrics like volume and surface area. They allow for the expression of mathematical operations using variables and constants. In our problem, algebraic expressions help us manage and simplify computations involving dimensions that might change.For example, in calculating the volume V = 8xyz, each of the variables x, y, and z represents a dimension that can vary. Algebraic expressions facilitate managing these variables without needing specific numeric values for each instance.Similarly, when dealing with surface area, we use the expression:\[ 16xy + 16xz + 2yz \]This highlights the power of algebraic expressions: they help condense complex calculations into manageable formats. By understanding how to manipulate these expressions—such as by factoring or expanding—you can solve or simplify various problems linked to the dimensions and properties of geometric shapes.