Problem 66 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(\mathscr{A}=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) for \(b_{2} \quad\) (Area of a trapezoid)

Step-by-Step Solution

Answer

\( b_2 = \frac{2\mathscr{A}}{h} - b_1 \)

Step 1: Identify the Formula and Given Variable

The formula given is for the area of a trapezoid: \[ \mathscr{A} = \frac{1}{2} h (b_1 + b_2) \]We are required to solve this equation for the variable \( b_2 \).

Step 2: Eliminate the Fraction

To get rid of the fraction, multiply both sides of the equation by 2.\[ 2\mathscr{A} = h (b_1 + b_2) \]

Step 3: Isolate the Bracketed Term

Divide both sides by \( h \) to isolate \( (b_1 + b_2) \):\[ \frac{2\mathscr{A}}{h} = b_1 + b_2 \]

Step 4: Solve for b_2

Subtract \( b_1 \) from both sides of the equation to solve for \( b_2 \):\[ b_2 = \frac{2\mathscr{A}}{h} - b_1 \]

Key Concepts

Understanding the Area of a Trapezoid

A trapezoid is a four-sided shape with at least one pair of parallel sides. These parallel sides are often referred to as the "bases" of the trapezoid. Knowing their lengths is crucial to solving area problems involving trapezoids. The area of a trapezoid can be thought of as the "space inside" it, similar to how you might think about coloring inside the lines. The formula to find this area is pretty straightforward: \[ \text{Area} = \frac{1}{2} h (b_1 + b_2) \]Here,

• \(h\) stands for the height of the trapezoid, which is the distance between the parallel sides.
• \(b_1\) and \(b_2\) are the lengths of those parallel sides.

This formula essentially calculates the average length of the two bases and then multiplies it by the height to find the area.

Manipulating Formulas

Manipulating formulas involves rearranging equations to make a different variable the subject. It's a crucial skill for solving algebra problems, as it allows you to express one variable in terms of others. In our example with the trapezoid area formula, we need to rearrange it to solve for \(b_2\). To start, identify what you want to solve for—in this case, \(b_2\). This step guides all subsequent actions in formula manipulation.

• Multiply both sides by 2 to eliminate the fraction. This step simplifies the equation and makes it easier to work with.
• Next, divide by \(h\) to isolate the term \((b_1 + b_2)\). You now have \(b_2\) grouped with \(b_1\), bringing you closer to having \(b_2\) on its own.
• The final step is to subtract \(b_1\) from both sides. This isolates \(b_2\) completely, allowing you to express it solely in terms of \(\mathscr{A}\), \(h\), and \(b_1\).

Always remember: each step in formula manipulation brings you closer to isolating the desired variable.

Solving for a Variable

When you solve for a variable, you essentially "untangle" the math to find what one element of an equation equals. This often involves reversing operations while maintaining balance on both sides of the equation, much like balancing a seesaw. In our trapezoid problem:

• Start with the modified equation: \(2\mathscr{A} = h (b_1 + b_2)\).
• Next, divide by \(h\) to simplify: \(\frac{2\mathscr{A}}{h} = b_1 + b_2\).
• Finally, subtract \(b_1\) to solve for \(b_2\): \(b_2 = \frac{2\mathscr{A}}{h} - b_1\). This process uses basic arithmetic operations to "solve" for \(b_2\), allowing you to substitute numbers for \(\mathscr{A}\), \(h\), and \(b_1\) to find a numeric solution.

Mastering this process requires practice, but it's deeply rewarding as it forms the foundation for much of algebra and beyond.