Problem 39 - Thinking Mathematically

Question

In Exercises 39-42, use an algebraic equation to determine each rectangle's dimensions. A rectangular field is four times as long as it is wide. If the perimeter of the field is 500 yards, what are the field's dimensions?

Step-by-Step Solution

Answer

The dimensions of the field are 50 yards wide and 200 yards long.

Step 1: Define the Variables

Let’s define the width of the field as \(x\) and the length as \(4x\). This is given by the problem stating that the rectangular field is four times as long as it is wide.

Step 2: Use the Perimeter Definition

The formula for the perimeter \(P\) of a rectangle is \(P = 2*(length + width)\). Substituting the given values, we get \(500 = 2*(4x + x)\).

Step 3: Solve the Equation

Opening up the brackets, we get \(500 = 2*5x\), or \(500 = 10x\). Solving for \(x\), we divide both sides by 10 to give us (width) \(x = 50\). To find the length, we substitute \(x=50\) into the length equation. Thus, the length is \(4*50 = 200\). Therefore, the field is 50 yards wide and 200 yards long.

Key Concepts

Algebraic Equations

Algebraic equations are at the heart of solving various mathematical problems, particularly when it comes to finding unknown values. An equation represents a balance between two expressions with an equal sign (). When we have a scenario where a rectangle's perimeter needs to be calculated, we use an algebraic equation to tie the different parts of the problem together.

It’s important to identify what we’re solving for, which in our exercise, is the rectangular dimensions given the perimeter. Encountering an equation like ) makes it seem straightforward — just simple multiplication and addition. However, the real skill comes into play when we substitute the values and solve for the variables, systematically moving towards the solution.

Familiarizing With Algebra

Understanding algebraic equations is crucial because it is a fundamental skill that you’ll use in multiple areas of mathematics and real-life applications. Algebra helps us describe relationships and changes, allowing us to predict and understand different scenarios. It’s a foundational tool that equips you to approach problems logically and find concrete solutions.

Rectangular Dimensions

Rectangular dimensions refer to the length and width of a rectangle, which are the fundamental components needed to determine the perimeter and area of the shape. In geometry, these dimensions are used to describe the size and shape of the rectangle. When we focus on the perimeter, we are essentially adding up all sides of the rectangle.

The formula for the perimeter of a rectangle is given by
), where the length and width are represented by distinct variables. Remember, understanding the relationship between these dimensions is key to solving for them. For instance, if a rectangle’s length is four times its width, as stated in the problem, this piece of information becomes a bridge — in form of a ratio — that connects the two dimensions.

The Importance of Ratios

Recognizing how dimensions relate to each other through ratios is essential in geometry. Ratios represent a comparative measure between quantities and in this case, simplify how we express the size of one side relative to another. This concept allows for more efficient problem-solving and helps visualize the problem.

Solving for Variables

Solving for variables is the process of finding the values of the symbols that represent the unknown in an algebraic equation. When dealing with rectangles, these variables typically stand for the dimensions of length and width. The goal is to manipulate the equation while adhering to mathematical rules, such as balancing both sides to isolate the variable.

In our exercise, solving for the variable begins once we've established the relationship between the width () and length (). With the perimeter known, we use algebraic manipulation — multiplying and dividing — to find . These steps are critical as they lead us to the values of and subsequently, the actual dimensions of the rectangle.

Methodical Approach

Methodically solving for follows a clear set of steps which, if applied properly, will always lead to the correct answer. This includes simplifying expressions, combining like terms, and finally, isolating the variable to find its value. Mastery of these steps will not only help you with geometric problems but also in more complex algebraic equations encountered in advanced mathematics.