Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 30 meters and the length is 3 meters greater than the width.

Step-by-Step Solution

Answer

The dimensions of the rectangle are 6 meters (width) and 9 meters (length).

Step 1: Setting up the equation

From the statement, we know that the perimeter of the rectangle is 30 meters and the length of the rectangle is 3 meters greater than the width. Suppose the width of the rectangle is \(w\) meters. Therefore, the length of the rectangle is \(w+3\) meters. We can write the equation for the perimeter of the rectangle as follows: \(2 \times (w +(w+3)) = 30\)

Step 2: Solving the equation

To solve above equation, first we distribute 2 inside the brackets, that gives us \(2w + 2(w + 3) = 30\), which simplifies to \(2w + 2w + 6 = 30\). Combine like terms to get \(4w + 6 = 30\). Next, subtract 6 from both sides of the equation to yield \(4w = 24\). Lastly, divide both sides of the equation by 4 to get \(w = 6\). So, the width of the rectangle is 6 meters.

Step 3: Find the length

From our initial statement, we know that the length is 3 meters longer than the width. Now that we have \(w = 6\), we simply add 3 to find the length: Length = \(w + 3 = 6 + 3 = 9\) meters. So, the length of the rectangle is 9 meters.

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around its edges. To calculate the perimeter, you need to know both the length and the width of the rectangle. The formula for the perimeter (P) of a rectangle is expressed as follows:

• \( P = 2 \times (\text{length} + \text{width}) \)

Here, you multiply the sum of the length and the width by 2 because a rectangle has two pairs of opposite sides that are equal.
In the given problem, the perimeter is specified to be 30 meters. Understanding how to set up the equation for the perimeter is crucial, as it helps you find the unknown dimensions of a rectangle when certain conditions like this are provided.

Width and Length Relationship

In some problems, the relationship between the width and the length of a rectangle is defined by a particular condition. For this exercise, the length is specified to be 3 meters greater than the width. This relationship can be expressed mathematically as:

• \( \text{length} = \text{width} + 3 \)

Using algebra to incorporate this relationship into the perimeter formula, you substitute \( \text{width} \) and \( \text{width} + 3 \) for the length in the perimeter equation.
This step is key in translating the problem into an equation that can be solved to find the dimensions of the rectangle. Recognizing such relationships helps simplify the problem and guides you in forming an accurate equation for solving the unknowns.

Rectangular Dimensions

Finding the dimensions of a rectangle involves solving for its length and width using provided conditions. In this problem, the equation \( 4w + 6 = 30 \) originates from substituting the width and its relationship with the length into the perimeter formula. This equation is solved in a series of simple steps:

• First, simplify the equation by combining like terms.
• Next, isolate the variable \(w\) by performing inverse operations - subtract to simplify, then divide to solve for \(w\).

Solving this gives the width as 6 meters.
Once the width (\(w\)) is found, computing the length is straightforward—just add 3 to the width according to the condition given, yielding a length of 9 meters. Understanding these calculation steps allows you to tackle similar problems effectively, knowing how each dimension relates to others.