Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the specified variable. \(S=2 L W+2 W H+2 H L\) for \(H \quad\) (Surface area of a rectangular box )

Step-by-Step Solution

Answer

\(H = \frac{S - 2LW}{2W + 2L}\)

Step 1: Understand the Equation

We start with the formula for the surface area of a rectangular box: \( S = 2LW + 2WH + 2HL \), where we are asked to solve for \( H \).

Step 2: Rearrange the Equation

To isolate \( H \), first subtract \( 2LW \) from both sides of the equation: \( S - 2LW = 2WH + 2HL \).

Step 3: Factor Out H

Notice that \( H \) appears twice on the right. Factor \( H \) out of the terms: \( S - 2LW = H(2W + 2L) \).

Step 4: Solve for H

To isolate \( H \), divide both sides by \( (2W + 2L) \): \( H = \frac{S - 2LW}{2W + 2L} \).

Key Concepts

Surface Area

Understanding how to calculate the surface area of a shape is essential in geometry. The surface area refers to the total area that the surface of an object occupies. For a three-dimensional shape like a rectangular box or prism, this involves calculating the area of every side and adding them up.

In the formula given for the surface area of a rectangular box, the equation is:

• \[ S = 2LW + 2WH + 2HL \]

Here:

• \( S \) is the surface area.
• \( L \), \( W \), and \( H \) stand for the length, width, and height respectively.

Each term in the formula corresponds to the area calculations for pairs of the three different faces:

• \( 2LW \) is the sum of the areas of the front and back faces,
• \( 2WH \) covers the areas of the left and right sides,
• \( 2HL \) accounts for the top and bottom faces.

Calculating surface area is practical for tasks such as determining the amount of material needed to cover a box entirely, like painting or wrapping.

Rectangular Box

A rectangular box, also known as a rectangular prism, is a three-dimensional figure with six faces, all of which are rectangles. This shape is ubiquitous in everyday life. Consider the cereal boxes in your kitchen or the gift boxes around holidays.

Characteristics of a rectangular box include:

• It has three pairs of parallel faces.
• Opposite faces are not only parallel but also are congruent, meaning they are the same in shape and size.
• It comprises vertices, edges, and faces. A standard rectangular box will have eight vertices, twelve edges, and six faces.

These properties make the rectangular box a simple and predictable shape, ideal for solving mathematical problems—especially those involving surface area and volume. As it uses perpendicular sides, the calculations involved generally rely on straightforward multiplication and addition.

Variable Isolation

Variable isolation is a fundamental technique in algebra. It involves rearranging an equation so that the variable of interest stands alone on one side of the equation. This concept is crucial when solving for a variable in terms of other quantities.

For the surface area equation for a rectangular box, the process of isolating \( H \) involves several steps:

• Begin by subtracting the terms not involving \( H \) from both sides of the equation. This removes any unrelated values from the side of the equation we are focusing on.
• Next, notice \( H \) appears as a common factor in the terms on the right-hand side. Factoring saves us time by turning multiple terms into a single expression.
• Finally, divide the remaining expression by the factor combined.

The main goal is to rewrite the equation efficiently to express \( H \) as:

• \[ H = \frac{S - 2LW}{2W + 2L} \]

Variable isolation allows you to clarify logical relationships between different quantities and is applicable in many areas beyond just geometry, such as physics and economics.