Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=\frac{1}{2} b h \text { for } h$$

Step-by-Step Solution

Answer

Solve for \(h\), \(h = \frac{2A}{b}\)

Step 1: Understanding the formula

Recognize the formula. \(A=\frac{1}{2}bh\) 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: Isolate \(h\)

In order to isolate \(h\), first multiply both sides of the equation by 2 to remove the fraction: \(2A = bh\)

Step 3: Further simplify the expression

Divide both sides of the equation by \(b\) to finally isolate \(h\): \(h = \frac{2A}{b}\)

Key Concepts

Area of a Triangle

The area of a triangle is a fundamental concept in geometry, important for solving various real-world problems. To find the area, you use the formula \(A = \frac{1}{2} b h\), where:

• \(A\) represents the area of the triangle.
• \(b\) is the length of the base of the triangle.
• \(h\) is the height, which is a perpendicular line drawn from the base to the opposite vertex.

This simple formula helps calculate how much space is enclosed within the triangle. Understanding this mathematic principle is crucial in both academic settings and practical applications like architecture or design. Remembering to multiply the base by the height and then halving it gives an efficient way to calculate the area.

Isolating Variables

Isolating variables is a core algebraic skill that helps in solving equations. In a situation where you know a formula and need to solve for a specific variable, isolating that variable is essential.Consider the formula for the area of a triangle, \(A = \frac{1}{2} b h\). If you are given the area and the base and need to find the height, you must rearrange the formula to solve for \(h\). This process involves:

• Multiplying both sides by 2 to eliminate the fraction: \(2A = bh\).
• Dividing both sides by \(b\) to isolate \(h\): \(h = \frac{2A}{b}\).

Being adept at isolating variables allows you to manipulate equations to find unknown values effectively.

Mathematical Equations

Mathematical equations are expressions composed of variables, numbers, and operations that assert the equality of two expressions. They are fundamental in expressing mathematical relationships and rules, and they often require manipulation to reveal unknown quantities.In our example, the equation \(A = \frac{1}{2} b h\) encapsulates a relationship between the area, base, and height of a triangle. To utilize this equation:

• Identify which variable you need information about.
• Use algebraic methods, like isolating variables, to rearrange the equation accordingly.

Equations are the backbone of problem-solving across mathematics and science. Understanding how to work with them offers insights into countless physical phenomena and processes, enabling us to mathematically describe the world around us.