Problem 57 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the given formula

The given formula is \(A=\frac{1}{2} b h\), which is the formula for the area of a triangle, where 'A' is the area, 'b' is the base and 'h' is the height.

Step 2: Rewrite the formula

To solve for 'b', we need to isolate 'b' on one side of the equation. We can multiply both sides of the equation by \(2\) to get rid of the denominator on the right side. This gives us \(2A = b h\).

Step 3: Solve for 'b'

To isolate 'b', we can divide both sides of the equation by 'h'. Hence, the formula for 'b' is given by \(b=\frac{2A}{h}\).

Key Concepts

Area of a Triangle

Understanding how to calculate the area of a triangle is fundamental in geometry. The area represents the amount of space confined within the triangle. If a triangle is presented to you, determining its area means finding out how much surface it covers.
The formula to calculate the area of a triangle is straightforward:

• The formula is given by \(A = \frac{1}{2} b h\), where:
  • \(A\) stands for the area of the triangle.
  • \(b\) represents the base of the triangle.
  • \(h\) is the height, which is the perpendicular distance from the base to the opposite vertex.

Knowing this formula allows you to explore other mathematical concepts. For instance, if one does not know the base or the height but knows the area and one of these variables, the formula can be adjusted to solve for the unknown variable. This often involves using techniques of algebra to rearrange the equation, which we will discuss in the following sections.

Formula Manipulation

Manipulating formulas is a key skill in algebra, which involves rearranging the equation to express a different variable. This skill is crucial when you need to solve real-world problems or translate general equations to fit into specific scenarios.
Using the area of a triangle formula \(A = \frac{1}{2} b h\), you can see the potential flexibility. If asked to solve for the base (\(b\)), the focus would shift from finding the area to isolating \(b\) on one side of the equation.

• Firstly, acknowledge the original form of the equation: \(A = \frac{1}{2} b h\).
• Multiply both sides by \(2\) to eliminate the fraction: \(2A = b h\).
• This step simplifies our equation into a more manageable expression for further manipulation. Ensuring every term is correctly transformed while maintaining the equation's balance is key.

Understanding formula manipulation empowers you to solve equations, compare proportional relationships, and comprehend how different factors in an equation affect each other.

Isolating Variables

Isolating a variable refers to the process of rearranging an equation so that the desired variable is alone on one side. This technique is essential for solving equations and finding particular values.
Continuing with the previous example, once you've reached \(2A = b h\), the next task is to isolate \(b\):

• To achieve this, divide both sides by \(h\), giving \(b = \frac{2A}{h}\).
• This step effectively leaves the base \(b\) isolated, now expressed in terms of the known quantities \(A\) (area) and \(h\) (height).
• The simplification ensures that \(b\)'s value can be easily calculated if you know the other quantities.

Isolating variables not only simplifies solving equations, but also strengthens the understanding of how different elements in mathematical expressions are interconnected. Practicing this technique builds foundational algebra skills that are applicable across various math and science fields.