Problem 27 - Elementary Algebra

Question

Use numerical evaluation on the equations. Geometry (perimeter of a rectangle) \(P=2 l+2 w . \) Find \(P\) if \(l=74\) and \(w=16\).

Step-by-Step Solution

Answer

Answer: The perimeter of the rectangle is 180 units.

Step 1: Write down the formula for the perimeter of a rectangle

Start by writing down the formula for the perimeter of a rectangle: \(P = 2l + 2w\).

Step 2: Substitute the given values for l and w

Next, plug in the given values for l and w into the formula. We are given \(l = 74\) and \(w = 16\). Replace l with 74 and w with 16 in the formula: \(P =2(74) + 2(16)\).

Step 3: Calculate the perimeter

Now, calculate the perimeter by following the order of operations. Multiply before adding: \(P = 2(74) + 2(16)\) \(P = 148 + 32\) Now add the two numbers: \(P = 180\) So, the perimeter of the rectangle is 180 units.

Key Concepts

Numerical Evaluation

Numerical evaluation is the process of substituting numbers for variables and performing the arithmetic to obtain a numerical result. Let's take a closer look at how this applies to finding the perimeter of a rectangle. With the given lengths, you need to evaluate by substituting these into the formula
Using our exercise, the first step is to recognize the formula for the perimeter of a rectangle, which is \(P = 2l + 2w\). You then replace the variables \(l\) and \(w\) with the actual measurements, so if \(l = 74\) units and \(w = 16\) units, you'd substitute these into the formula to get \(P = 2(74) + 2(16)\).
Once the substitution is made, the arithmetic part of numerical evaluation comes into play. You'd multiply and add the values as the order of operations dictates, ultimately arriving at a final numerical answer for the perimeter, which represents the total distance around the rectangle.

Geometric Formulas

Geometric formulas are essential tools for solving various geometry problems. They are the set rules that tell you how to calculate different properties of shapes, like area, perimeter, volume, and more. The formula for the perimeter of a rectangle is a fundamental example, expressed as \(P = 2l + 2w\).

Understanding the Rectangle Perimeter Formula

The formula combines the lengths of all four sides. Because a rectangle has opposite sides that are equal, the formula simplifies the calculation by only needing the length \(l\) and the width \(w\) of the rectangle. The perimeter is then calculated by adding the lengths of all four sides, which leads to doubling the length and doubling the width before adding them together. Remembering the correct geometric formula is key to accurately solving the problem.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is a set of rules that determines in which order you should solve different parts of a mathematical expression. In our context, after substituting the values for \(l\) and \(w\) in the perimeter formula \(P = 2l + 2w\), we must follow the order of operations to accurately find its numerical value.
In our case, multiplication comes before addition, following PEMDAS. So we first multiply 2 by 74 and 2 by 16, then add the resulting figures (148 and 32, respectively) to find the perimeter. Hence, the order in which we execute the operations is crucial to arriving at the correct answer of 180 units for the perimeter.