Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

A rectangular field is five times as long as it is wide. If the perimeter of the field is 288 yards, what are the field's dimensions

Step-by-Step Solution

Answer

The dimensions of the field are 24 yards by 120 yards.

Step 1: Define the Variables

Let the width of the rectangle be \(x\) yards. The length of the rectangle is given in the problem as five times the width. Therefore, we can define the length as \(5x\) yards.

Step 2: Write an Equation for Perimeter

The perimeter \(P\) of a rectangle is given by the formula \(P = 2L + 2W\), where \(L\) is length and \(W\) is width. Using the defined variables, this equation becomes \(288 = 2(5x) + 2(x) = 12x\).

Step 3: Solve the Equation for \(x\)

Now we will solve the equation for \(x\). Divide both sides of the equation by 12 to get \(x = 288/12 = 24\) yards.

Step 4: Find the Dimensions of the Field

The width we have found is \(x = 24\) yards. Multiply that by 5 to find the length, \(5x = 5(24) = 120\) yards. Therefore, the field's dimensions are 24 yards by 120 yards.

Key Concepts

Perimeter of a Rectangle

Understanding the perimeter of a rectangle is fundamental in solving problems related to rectangular shapes. The perimeter is the total distance around the outside of a rectangle and it can be found using the formula:
\[ P = 2L + 2W \]
where \( P \) represents the perimeter, \( L \) is the length, and \( W \) is the width of the rectangle. In practical scenarios, such as in our exercise about a rectangular field, calculating the perimeter allows us to understand the fencing needed or, as in the exercise, to reverse-calculate dimensions given the total distance around the field.

Algebraic Equation Solving

Solving algebraic equations is like unraveling a puzzle. The key is to perform operations that will isolate the variable (or unknown quantity) you're solving for. In our example, after setting up the equation for perimeter
\[ 288 = 2(5x) + 2(x) \]
we combined like terms and divided each side by the same number, 12, in this case, to solve for \( x \). It is crucial to maintain the balance of the equation, meaning whatever you do to one side, you do to the other. This ensures the equation's integrity and helps you reach the correct solution.

Rectangular Field Problem

The rectangular field problem is a common type of question that applies algebraic concepts to real-world scenarios. Here, you're often given specifics about the dimensions of a field and asked to find missing measurements. This type of problem not only tests your algebra skills but also your ability to translate a word problem into a mathematical model. By carefully reading the problem, defining variables, and applying the perimeter formula, you can figure out unknown dimensions, as we did to find that the field's dimensions are 24 yards wide and 120 yards long.

Defining Variables in Algebra

In algebra, defining variables is a critical step in translating word problems into equations. A variable is a symbol, usually a letter, that represents a number that can change or vary. In our exercise, we let the width be represented by \( x \). Then, we defined the length in terms of this variable (\( 5x \)) since the problem stated it was five times as long as the width. By expressing all parts of the problem with variables, we create an algebraic representation that can be manipulated to solve for these unknown quantities.