Problem 24 - Elementary Algebra

Question

Use numerical evaluation on the equations. Geometry (area of a parallelogram) \(A=b h . \) Find \(A\) if \(b=16\) and \(h=6\).

Step-by-Step Solution

Answer

Answer: The area of the parallelogram is 96 square units.

Step 1: Identify the given values

We are given the base (b) as 16 and height (h) as 6.

Step 2: Apply the formula for the area of a parallelogram

Using the formula \(A = b * h\), we can substitute the given values into the equation to find the area (A).

Step 3: Calculate the area

With the given values, the formula for the area becomes: \(A = 16 * 6\). To find the area, multiply the base (16) by the height (6): \(A = 16 * 6 = 96\)

Step 4: Write the final answer

The area (A) of the parallelogram is 96 square units.

Key Concepts

Numerical Evaluation

When faced with a geometric problem, numerical evaluation is a straightforward technique used to find the value of a given expression by substituting numbers for variables. In the context of finding the area of a parallelogram, we are given specific measures for the base and height. By replacing these numerical values into the area formula—rather than working with abstract variables—we obtain tangible results that represent real-world measurements. To prevent errors during the evaluation, it's important to carefully insert each given number into its correct place within the algebraic expression, as demonstrated in the provided exercise.

Geometric Formulas

Geometric formulas are essential tools for solving problems in geometry. These formulas relate various geometric figures' dimensions to their properties, such as area or perimeter. The area of a parallelogram is found using the formula A = b × h, where A represents the area, b is the base, and h is the height. It is critical to comprehend that the height is the perpendicular distance from the base to the opposite side. Recognizing the appropriate formula for each shape is key in geometry, as it serves as a basis for developing problem-solving strategies and understanding the properties of shapes.

Algebraic Expressions

In geometry, algebraic expressions are often used to generalize the relationships between different parts of geometric figures. An expression can include variables (letters representing unknown values), constants (known numbers), and arithmetic operations. The area formula for a parallelogram, A = b × h, is such an expression. While it can be applied to any parallelogram by substituting the specific values for base and height, it remains a powerful general formula due to its ability to accommodate different numerical values. It is this generality that makes algebraic expressions so widely applicable in various geometric contexts.

Geometry

Geometry is a branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. It allows us to understand the physical world by abstracting the space and shapes around us. To solve geometric problems, such as finding the area of a parallelogram, one must understand both the specific properties of the shape and also the general principles of spatial reasoning. By seamlessly connecting algebraic expressions and geometric formulas to represent physical space quantitatively, geometry serves as a fundamental tool for various fields, including architecture, engineering, art, and natural sciences.