Problem 67 - A Graphical Approach to College Algebra

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

The solution for \(H\) is \(H = \frac{S - 2LW}{2(W + L)}\).

Step 1: Isolate Terms Involving H

To solve for \(H\), start by isolating the terms that include \(H\). The original formula is \(S = 2LW + 2WH + 2HL\). We need to move the \(2LW\) term to the other side of the equation: \(S - 2LW = 2WH + 2HL\).

Step 2: Factor Out H

To make the equation easier to solve, factor \(H\) out of the terms on the right-hand side. This gives you: \(S - 2LW = H(2W + 2L)\).

Step 3: Solve for H

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

Step 4: Simplify the Expression

Finally, further simplify the denominator \(2W + 2L\), which can also be written as \(2(W + L)\). Therefore, \(H = \frac{S - 2LW}{2(W + L)}\).

Key Concepts

Solving Equations

When solving equations, we aim to find the value of an unknown variable that makes the equation true. Equations often have variables, such as letters or symbols, that represent unknown values. In algebra, solving an equation typically involves manipulating the equation so that the variable is isolated on one side. This makes it easier to determine its value.
To successfully solve equations, we follow a methodical process:

• Look at the equation and identify the terms that need to be moved or combined.
• Use inverse operations, like subtraction for addition, to eliminate terms from one side.
• Continue to simplify the equation until the variable is by itself.

Understanding each of these steps allows for accurate problem-solving and lays the groundwork for more complex algebraic tasks. In our example, solving meant isolating the variable "H" from the surface area equation.

Surface Area

Surface area refers to the total area that the surface of an object occupies. For a three-dimensional shape like a rectangular box, calculating surface area involves adding up the areas of all its faces. Each face of the box is a rectangle, and the formula for the surface area is:\[ S = 2LW + 2WH + 2HL \]where

• \(L\) is the length,
• \(W\) is the width, and
• \(H\) is the height.

To solve for one of these dimensions, such as height \(H\), you need to isolate that variable. Understanding how to manipulate this formula is crucial for problems involving dimensions in real-life applications, like packaging, construction, and design.

Rectangular Box

A rectangular box, also known as a rectangular prism, is a three-dimensional geometric shape. It has six faces, all of which are rectangles, with opposite faces being equal in size. This shape can be described by its length (\(L\)), width (\(W\)), and height (\(H\)).
Rectangular boxes are common in everyday life, from cereal boxes to shipping containers.
Understanding the properties of a rectangular box helps in determining important measures such as volume and surface area. Calculating these measures involves formulas that include all three dimensions: length, width, and height. Rectangular boxes are key in many geometry problems, significantly aiding in space management and optimization tasks.

Variable Isolation

Variable isolation is a significant concept in algebra. It involves the process of rearranging an equation so that the variable of interest is alone on one side of the equation. This is essential for solving equations and finding the value of the unknown.
In the context of the surface area equation for a rectangular box, isolating \(H\) required several steps:

• First, terms not involving \(H\) were moved to the other side.
• Next, \(H\) was factored out from the remaining terms.
• Finally, the equation was manipulated so that \(H\) stood alone.

This method simplifies the equation, allowing for a clear solution for \(H\). Becoming proficient with variable isolation empowers students to tackle a wide range of mathematical problems with confidence.