Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

A rectangular field is four times as long as it is wide. If a perimeter of the field is 500 yards, what are the field's dimensions?

Step-by-Step Solution

Answer

The width of the field is 50 yards and the length of the field is 200 yards.

Step 1: Define the variables

Let \(x\) be the width of the field and \(4x\) be the length because the length is four times the width.

Step 2: Understand the problem in terms of the variables

The problem involves a rectangle and its perimeter. The formula for the perimeter of a rectangle is \(2*(length + width)\). Since we define the length as \(4x\) and the width as \(x\), we can substitute these into the formula and set it equal to 500 (the given perimeter): \(2*(4x + x) = 500\).

Step 3: Simplifying the equation

Simplify the equation: \(2*5x = 500\), which turns into \(10x = 500\).

Step 4: Solve for \(x\)

Now solve for \(x\) by dividing both sides of the equation by 10: \(x = 500 / 10\).

Step 5: Calculate the dimensions

So \(x = 50\) yards is the width of the field and the length would be \(4x = 4*50 = 200\) yards.

Key Concepts

Rectangular field

When tackling problems involving a rectangular field, it's useful to visualize the shape. A rectangular field has two sets of equal sides. The longer sides are known as the length, while the shorter ones are the width. In this particular problem, it is stated that the length is four times the width, establishing a clear relationship between the two dimensions. Visualizing a rectangle where the length is significantly longer than the width can help students understand how these dimensions relate to each other. It highlights the importance of geometrical relationships in real-world contexts like farming and architecture.

Perimeter formula

The perimeter of a rectangle is the total distance around it. By definition, it is calculated by adding the length and width and then multiplying the sum by two. The formula is expressed as:

• \( P = 2 imes (l + w) \)

In our exercise, the given perimeter is 500 yards, and the length and the width are expressed in terms of the same variable. Making use of the perimeter formula, we can substitute these expressions into the equation \(2*(4x + x) = 500\). The formula is crucial when you want to find missing dimensions if you know the perimeter.

Solving equations

Solving equations involves finding the value of unknown variables that satisfy the equation. In our problem, once the perimeter formula is set up with variables, the next step is to simplify the equation. This involves straightforward arithmetic operations like addition and multiplication. In this case, the operation simplifies from \(2*(4x + x) = 500\) to \(10x = 500\). Such simplification is essential because it reduces complex equations to simpler forms, making it easier to solve for unknowns. Dividing both sides by 10 gives us the width \(x = 50\) yards.

Variable substitution

Variable substitution is used to turn verbal relationships and conditions into algebraic language that can be manipulated mathematically. To solve our problem, we first defined the width as \(x\). Since the length is four times the width, we labeled it as \(4x\). Substituting these expressions into the perimeter formula allowed us to create an equation based entirely on a single variable, which we then solved for the width. Understanding this concept ensures that students can convert complex real-world situations into simple, solvable mathematical problems.