Problem 74 - College Algebra

Question

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=2 l w+2 l h+2 w h\) for \(h\)

Step-by-Step Solution

Answer

The formula solved for \(h\) is \(h = \frac{A - 2 l w} {2l + 2w}\)

Step 1: Identify the term with the variable \'h\'

The expression containing \(h\) is \(2 l h + 2 w h\). We need this term to be isolated on one side.'

Step 2: Remove the terms without \'h\'

To isolate the term with \(h\), subtract \(2 l w\) from both sides of the equation. This gives us:\(A - 2 l w = 2 l h + 2 w h\)

Step 3: Factor the variable \'h\'

To solve for \(h\), we need it by itself. We can do this by factoring \(h\) out from the right side of the equation so that it only appears once. This gives us:\(A -2 l w = h (2l + 2w)\)

Step 4: Solve for \'h\'

Finally, to solve for \(h\), divide both sides of the equation by \(2l + 2w\). This gives us:\(h = \frac{A - 2 l w} {2l + 2w}\)

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in mathematics, particularly when it comes to solving equations. It involves applying various mathematical operations, such as addition, subtraction, multiplication, or division, alongside the properties of equality to transform an equation into a more workable or simplified form.

For instance, when given an equation like the exercise above, the steps involve manipulating algebraic expressions to isolate the term containing the variable of interest. Subtracting terms from both sides or factoring to consolidate variables are examples of algebraic manipulation. It's like solving a puzzle, where each move is calculated to get you one step closer to the solution - finding the value of the variable.

Isolate Variable

To isolate a variable means to manipulate an equation in order to get the variable by itself on one side of the equation. The operation's goal is to end up with an expression such as 'variable = some expression', which indicates the variable's value in terms of the other quantities involved.

In the given exercise, isolating the variable 'h' required moving other terms that did not contain 'h' to the other side of the equation. This process of isolation allows you to solve for a specific variable within an equation, which is particularly helpful in subjects like physics and engineering, where formulas often need to be rearranged to find an unknown quantity.

Factoring in Algebra

Factoring is another vital concept in algebra, which simplifies expressions and is key to solving equations efficiently. It involves breaking down a complex expression into a product of simpler factors. When a term appears in multiple parts of an expression, as 'h' appears in the equation from our exercise, factoring it can be a powerful tool.

Factoring allows you to pull out the common variable from each term it appears in, just as you would extract a thread woven into a tapestry. In our exercise, by factoring 'h' from the terms '2lh' and '2wh', we're able to group them into a single term, making it easier to then isolate 'h' and solve the equation.

Surface Area of a Rectangular Prism

Understanding the surface area of a rectangular prism is vital in fields like architecture, engineering, and design. The formula used in the exercise, \( A=2lw + 2lh + 2wh \), represents the total area of all six faces of a rectangular prism.

It's essential to visualize that a rectangular prism has three dimensions - length (l), width (w), and height (h). The formula accounts for the areas of the sides: two with dimensions 'l' by 'w', two with 'l' by 'h', and the last two with 'w' by 'h'. When we talk about solving for 'h', we're reshaping this formula to express the height in terms of the surface area and the other dimensions, providing valuable insights into the relationship between these geometric properties.