Problem 24 - Algebra for College Students

Question

$$ A=\frac{1}{2} h\left(b_{1}+b_{2}\right) \quad \text { for } h $$

Step-by-Step Solution

Answer

The value of \( h \) is \( \frac{2A}{b_1 + b_2} \).

Step 1: Understand the Problem

The given formula is the area formula for a trapezoid: \( A = \frac{1}{2} h(b_1 + b_2) \). We need to solve this equation for \( h \), which means isolating \( h \) on one side of the equation.

Step 2: Eliminate the Fraction

Since the equation involves a fraction, the first step is to eliminate it by multiplying both sides by 2. This gives us: \( 2A = h(b_1 + b_2) \).

Step 3: Isolate h

To solve for \( h \), divide both sides of the equation by \( (b_1 + b_2) \). This results in \( h = \frac{2A}{b_1 + b_2} \).

Key Concepts

Area of Trapezoid

The area of a trapezoid can be determined using a specific formula, which is quite helpful in geometry problems. A trapezoid is a four-sided figure with at least one pair of parallel sides. The formula to find the area is given by \[ A = \frac{1}{2} h (b_1 + b_2) \] where:

• \( A \) is the area of the trapezoid,
• \( h \) is the height of the trapezoid,
• \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides (bases) of the trapezoid.

This formula essentially calculates the average of the two bases, multiplied by the height, to provide the total space the trapezoid occupies on a flat surface. Understanding this formula is crucial for tackling problems related to areas in geometry, especially those involving irregular shapes like trapezoids.

Equation Solving

Solving equations is a fundamental aspect of algebra. An equation represents a balance between two expressions. To solve for a particular variable, such as in the trapezoid area problem, typically involves isolating the variable on one side. Let's look at how this is done:- Start with the original equation: \( A = \frac{1}{2} h (b_1 + b_2) \).- To eliminate the fraction, multiply both sides by 2, resulting in \( 2A = h (b_1 + b_2) \).- Next, to solve for \( h \), divide both sides by \( (b_1 + b_2) \), yielding \( h = \frac{2A}{b_1 + b_2} \).These steps showcase simple algebraic manipulation techniques, crucial for rearranging formulas to solve for particular variables. Practicing these steps enhances problem-solving skills, making it easier to handle complex algebraic expressions.

Geometry Formula

Geometry often requires using formulas to understand and calculate various properties of shapes. These formulas help determine lengths, areas, volumes, and other characteristics by relating different dimensions of the shape. In the context of trapezoids, the formula\( A = \frac{1}{2} h (b_1 + b_2) \) serves to compute the area using height and the sum of the bases.Key points to remember about geometry formulas:

• They translate physical properties into numbers and equations.
• They provide systematic approaches to finding unknown measures.
• Key to solving problems involving spatial understanding and measurement.

By mastering these formulas, you lay the groundwork for diving deeper into various aspects of geometry, from basic shapes to complex structures.