Problem 42 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ A=\frac{1}{2} b h \text { for } b $$

Step-by-Step Solution

Answer

The base \( b \) is \( b = \frac{2A}{h} \).

Step 1: Identify the Given Formula

We are given the formula for the area of a triangle: \( A = \frac{1}{2} b h \). In this formula, \( A \) is the area, \( b \) is the base, and \( h \) is the height of the triangle. We are tasked with solving for \( b \).

Step 2: Rearrange the Formula to Isolate b

To solve for \( b \), we need to isolate it on one side of the equation. Start by eliminating the fraction by multiplying both sides of the equation by 2: \( 2A = b h \).

Step 3: Solve for b

Now that we have \( 2A = b h \), divide both sides by \( h \) to get \( b \) alone: \( b = \frac{2A}{h} \). This simplifies the formula to express \( b \) in terms of \( A \) and \( h \).

Key Concepts

Rearranging Formulas

Understanding how to rearrange formulas is a crucial skill in mathematics that helps us find unknown values in expressions. When given a formula, you might be asked to solve for a different variable than what's typically solved for, such as finding the base of a triangle when its area and height are known.
This involves algebraic manipulation, where you change the structure of the formula without changing its meaning. A good way to approach this is by looking for opportunities to use operations such as addition, subtraction, multiplication, and division to move terms around.
Here’s the basic process:

• Focus on the variable you need to solve for, keep track of operations applied to that variable.
• Use inverse operations to move other terms to the opposite side of the equation.
• Ensure the variable you are solving for ends up by itself on one side of the equal sign.

Rearranging formulas is especially helpful in science and engineering where different variables are interrelated, and knowing how to isolate one can provide practical solutions to real-world problems.

Triangle Area Formula

The formula for the area of a triangle plays a fundamental role in both geometry and algebra. Understanding this formula can enhance how you solve problems involving triangles.
The triangle area formula is given by \[ A = \frac{1}{2} b h \] where \( A \) represents the area, \( b \) is the base, and \( h \) is the height. This formula tells us that the area of a triangle is half the product of its base and height.
The reason behind the formula is related to how much space a triangle occupies within a bounding rectangle where the base is one side, and the height is its perpendicular distance. When rearranging this formula to solve for a specific variable, understanding the concept of area will help maintain correct logical flow.

Isolating Variables

Isolating variables is an essential part of solving equations, allowing us to express one variable in terms of others given a known relationship.
In our exercise, isolating \( b \) means rewriting the equation so that \( b \) is by itself on one side. This involves elementary algebraic operations, where each step is about maintaining equation balance while making \( b \) the subject.
Here’s a quick rundown:

• You begin by transforming the equation to eliminate fractions or constants attached to the desired variable by performing inverse operations.
• Multiplying or dividing both sides by a common factor or the coefficient of the variable of interest.
• Ensuring all rearrangements keep the equation's integrity intact by maintaining equal operations on both sides.

Mastering the art of isolating variables not only assists in mathematical problem-solving but is also widely applicable in real-world scenarios where converting complex formulas into usable forms is required.