Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(A=\frac{1}{2} b h\) for \(h\)

Step-by-Step Solution

Answer

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

Step 1: Understand the formula

The formula given is for the area of a triangle: \( A = \frac{1}{2} b h \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height of the triangle. We need to rearrange this formula to solve for \( h \).

Step 2: Clear the fraction

To remove the fraction, we can multiply both sides of the equation by 2, which gives us: \( 2A = b h \).

Step 3: Solve for \( h \)

To isolate \( h \), divide both sides of the equation by \( b \): \( h = \frac{2A}{b} \).

Step 4: Double-check the rearranged formula

Revisit the rearranged formula \( h = \frac{2A}{b} \) to ensure it makes sense. This equation shows that the height \( h \) is twice the area \( A \) divided by the base \( b \).

Key Concepts

Area of a Triangle

Understanding the area of a triangle is foundational in geometry. The area is a measure of the surface a triangle covers on a plane. To calculate it, you can use the formula \( A = \frac{1}{2} b h \), where:

• \( A \) stands for the area of the triangle,
• \( b \) is the base, and
• \( h \) is the height.

The base can be any one side of the triangle, but the height must always be the perpendicular distance from that base to the opposite vertex. This formula works because a triangle is essentially half of a parallelogram, so the area is half the product of the base and height.

Remember, understanding this formula helps in many real-life applications, such as calculating land areas or understanding architectural designs.

Rearranging Formulas

Rearranging formulas is a useful mathematical skill that lets you solve for a specific variable in an equation. When you rearrange a formula, you're essentially expressing one variable in terms of others. For the triangle area formula, we started with \( A = \frac{1}{2} b h \) and needed to isolate \( h \).

Here are the steps to rearrange the formula:

• Multiply both sides by 2 to eliminate the fraction: \( 2A = b h \).
• Divide both sides by \( b \) to solve for \( h \): \( h = \frac{2A}{b} \).

This rearrangement tells us how to calculate the height when you know the area and base of a triangle. Rearranging formulas can also reveal how changing one part affects the others, which is invaluable in modeling and problem-solving across different areas.

Geometry Variables

Geometry involves different variables and how they relate to shapes and sizes. In the context of triangles:

• The base \( b \) is one side of the triangle, usually the bottom, but can be any side you're considering.
• The height \( h \) is a line perpendicular to the base, extending to the opposite vertex.
• Area \( A \), as we've learned, represents the space within the triangle.

These variables are not just numbers—they represent physical properties that change how the shape behaves.

Recognizing how these variables interact and impact mathematical formulas is crucial. For instance, a change in base or height directly alters the area, demonstrating the interconnectedness of these geometry variables.