Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate \(A=\frac{1}{2} h(a+b)\) for \(a=10, b=16,\) and \(h=7\)

Step-by-Step Solution

Answer

The area of the trapezoid is 91 square units.

Step 1: Identify the formula

In this problem, the formula of the area \(A\) of a trapezoid is given as \(A=\frac{1}{2} h(a+b)\).

Step 2: Substitute the given values

Substitute the given values into the formula. Therefore, \(A=\frac{1}{2} * 7(10+16)\).

Step 3: Simplify the expression within the brackets

Combine the two numbers inside the parenthesis (\(10+16\)) to simplify the expression. This results in \(A=\frac{1}{2} * 7 * 26\).

Step 4: Multiplication

Multiply the numbers to solve the expression. This results in \(A= 91\) square units.

Key Concepts

Algebraic Expression Evaluation

Understanding how to evaluate algebraic expressions is fundamental in solving mathematical problems. It essentially means finding the value of the expression when numbers are substituted for variables. Take, for instance, the formula for calculating the area of a trapezoid: \(A=\frac{1}{2} h(a+b)\). Here, \(A\) represents the area, \(h\) the height, and \(a\) and \(b\) the lengths of the two parallel sides. To evaluate this expression, one must replace each variable with its numerical value.

For our exercise, the given values are \(a=10\), \(b=16\), and \(h=7\). Evaluation involves substituting these values into the formula and solving for \(A\), as shown in the provided steps. It's like following a recipe where the ingredients (values) are mixed (substituted) into the formula (recipe) to yield the delicious result (solution). Remember, always perform operations according to the Order of Operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, parentheses come first, followed by multiplication.

Substitution in Algebra

Substitution is a method used in algebra to replace a variable with its corresponding value. It's like giving someone a placeholder for their coat check — you give something, and you get back exactly what belongs in that place. In the context of our area calculation, substitution is step 2 in the solution process.

Once the values for \(a\), \(b\), and \(h\) are known, they are systematically plugged into the formula. For the trapezoid area, this is where we replace \(a\) with 10, \(b\) with 16, and \(h\) with 7. The key is to insert these values carefully and accurately to avoid any mistakes that might lead to an incorrect answer. The concept holds true for all algebraic expressions beyond just geometric formulas. Mastering substitution is not only helpful for homework but also for applying algebra to real-world problems.

Simplification of Expressions

Simplification of expressions involves reducing an algebraic expression to its simplest form without changing its value. The goal is to make complex equations more manageable and understandable. Think of it as cleaning and organizing a messy room; once clean, it's much easier to navigate and find what you need. In the exercise, simplification comes into play in step 3, where we combine the sum inside the parentheses (\(10+16\)) to make the equation less cumbersome.

Afterward, you proceed to step 4, which is to multiply the simplified terms to reach the final result. Simplification often involves combining like terms, applying distributive properties, and reducing fractions. It's a vital skill in algebra that can simplify not only numbers but also the entire approach toward solving a variety of mathematical problems.