Problem 13 - Introductory Algebra for College Students

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2} b h\) for \(b\)

Step-by-Step Solution

Answer

The formula for \(b\) is \(b= \frac{2A}{h}\)

Step 1: Identify the equation

The equation is \(A=\frac{1}{2} b h\), which is the formula for a triangle's area in terms of its base \(b\) and height \(h\). The task is to isolate \(b\).

Step 2: Isolate b

To solve the formula for \(b\), you need to remove the fraction by multiplying both sides by \(\frac{2}{h}\). So, it'll give: \(b= \frac{2A}{h}\)

Step 3: Simplify

The equation is simplified into \(b= \frac{2A}{h}\). This is the representation of the base length in terms of the area and height of the triangle.

Key Concepts

Triangle Area Formula

When we talk about the triangle area formula, we're discussing a method to find the amount of space within the borders of a triangle. The formula is given as \(A = \frac{1}{2} \, b \, h\), where \(A\) represents the area, \(b\) is the base of the triangle, and \(h\) is the height. This formula is widely applied because triangles are fundamental shapes in geometry and other mathematical applications.
- **Why divide by 2?** The "\(\frac{1}{2}\)" in this formula is crucial because a triangle is essentially half of a rectangle if you consider the base and height the same way. Thus, dividing by 2 accounts for this fact, ensuring the calculation is accurate.
- **Units of measurement:** Generally, the area is measured in square units relative to the units used for base and height. For example, if base and height are in meters, the area will be in square meters.
This formula is straightforward and highly useful for various applications, from simple calculations to complex engineering problems.

Isolating Variables

Isolating variables is a fundamental algebra skill that helps in finding the value of a particular variable when it is part of an equation. This often involves rearranging the equation so that the variable we are solving for stands alone on one side of the equation.
To isolate a variable, follow these steps:

• Identify the variable that needs to be isolated. In our exercise, it's the base \(b\).
• Perform inverse operations to cancel out other terms or coefficients from around the variable. For example, when isolating \(b\) in \(A = \frac{1}{2} \, b \, h\), we multiplied both sides by \(\frac{2}{h}\) to undo the division.

Remember, whatever operation you do to one side of an equation, you must also do to the other side to maintain balance. This practice ensures that the equation remains equivalent during the transformation process.

Algebraic Manipulation

Algebraic manipulation is a technique used to rewrite equations or expressions, often to solve for a specific variable. In our example, we started with \(A = \frac{1}{2} \, b \, h\) and through manipulation, solved for \(b\). Key steps in algebraic manipulation include:

• Understanding the equation and knowing the goal of manipulation. Here, the goal was to express \(b\) in terms of \(A\) and \(h\).
• Applying mathematical operations systematically, such as the multiplication we used to eliminate fractions.
• Simplifying the expression where possible to achieve the simplest form of the desired variable, which resulted in \(b = \frac{2A}{h}\).

These manipulations are not just about solving for a variable but also about simplifying complex expressions, making them easier to understand and work with.