Problem 38 - College Algebra

Question

The area of a trapezoid is given by \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) Use the formula to find the area of a trapezoid with \(h=6, b_{1}=14,\) and \(b_{2}=8\)

Step-by-Step Solution

Answer

The area of the trapezoid is 66.

Step 1: Understanding the Formula

The formula for the area of a trapezoid is given by \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\). In this formula, \(A\) represents the area, \(h\) is the height, and \(b_{1}\) and \(b_{2}\) represent the lengths of the two parallel sides (bases) of the trapezoid.

Step 2: Plugging in Known Values

Substitute the given values into the formula. The height \(h\) is 6, base \(b_{1}\) is 14, and base \(b_{2}\) is 8. Thus, the formula becomes: \[ A = \frac{1}{2} \times 6 \times (14 + 8) \]

Step 3: Simplifying Inside the Parentheses

First, calculate the sum of \(b_{1}\) and \(b_{2}\): \[ 14 + 8 = 22 \]So, the equation now is: \[ A = \frac{1}{2} \times 6 \times 22 \]

Step 4: Calculating the Area

Now, multiply the values: 1. First, multiply the height by the sum of the bases: \[ 6 \times 22 = 132 \]2. Then, apply the \(\frac{1}{2}\) factor: \[ \frac{1}{2} \times 132 = 66 \]Thus, the area \(A\) of the trapezoid is 66.

Key Concepts

Geometry

Geometry is a fascinating branch of mathematics that deals with shapes, sizes, and the properties of space. It helps us understand the world around us by exploring the dimensions of various objects.
One of the key aspects of geometry is understanding different geometric shapes, such as triangles, squares, circles, and trapezoids. Each shape has its unique properties and formulas that help calculate dimensions like area and perimeter.
Let's take a closer look into trapezoids, a specific type of quadrilateral that plays a significant role in geometry.

Trapezoid Properties

A trapezoid is a unique quadrilateral that has one pair of parallel sides. These parallel sides are referred to as the bases of the trapezoid. The two other sides, which are not parallel, are called the legs.
The height of a trapezoid is the perpendicular distance between the two bases. Understanding this setup is crucial for calculating the area.
In some contexts, especially in different countries, trapezoids are also called trapeziums. Regardless of naming, the essential properties remain the same:

• Exactly one pair of parallel sides.
• The height connects the two bases at right angles.
• The sum of angles in a trapezoid is always 360 degrees.

These properties are foundational to using the trapezoid area formula effectively.

Mathematical Formulas

Mathematical formulas are powerful tools that help us find solutions to various problems. They give us a structured way to calculate lengths, areas, and other measurements.
For trapezoids, the area formula is especially useful. The formula for the area of a trapezoid is: \[ A = \frac{1}{2} h \left(b_1 + b_2\right) \]where:

• \(A\) stands for the area of the trapezoid.
• \(h\) is the height (perpendicular distance between bases).
• \(b_1\) and \(b_2\) are the lengths of the parallel sides (bases).

By substituting these values into the formula, you can determine the specific area of the trapezoid. This process involves basic arithmetic operations such as addition and multiplication, making it accessible even for beginners. Understanding and applying such formulas regularly is a step towards mastering mathematical problem-solving skills.