Solving for the Dimensions of a Rectangle Given its Area and Relationship Between Length and Breadth

In this article, we explore solving for the dimensions of a rectangle given its area and the relationship between its length and breadth. We will walk through a series of examples and provide step-by-step solutions to find the length and breadth of the rectangle.

Example Problem: Rectangle with Area 180m2 and Breadth 3m Less than Length

The problem statement tells us that the breadth of a rectangle is 3 meters less than its length. We are also aware that the area of the rectangle is 1802. To find the dimensions, we will use the quadratic equation method. Let's break down the steps:

Step 1: Define Variables

Let the length of the rectangle be L meters.

According to the problem, the breadth B is 3 meters less than the length. Therefore, we can express the breadth as:

B L - 3

Step 2: Set Up the Area Equation

The area A of the rectangle is given by the formula:

A L × B

We know from the problem statement that the area is 1802. Substituting the expression for breadth into the area formula:

180 L × (L - 3)

Step 3: Form a Quadratic Equation

Expanding and rearranging to form a standard quadratic equation:

180 L^2 - 3L

L^2 - 3L - 180 0

Step 4: Solve the Quadratic Equation

To solve the quadratic equation, we use the quadratic formula:

L ( -b ± √(b 2 - 4ac) ) / (2a)

Where a 1, b -3, and c -180.

First, calculate the discriminant:

b^2 - 4ac (-3)^2 - 4 × 1 × (-180) 9 720 729

Applying the quadratic formula:

L (3 ± √729) / 2 (3 ± 27) / 2

This gives us two potential solutions:

L (30 / 2) 15

L (-24 / 2) -12

Since the length cannot be negative, we discard L -12. Therefore, the length L is 15 meters.

Step 5: Find the Breadth

Using the relationship between the length and breadth:

B L - 3 15 - 3 12 meters

Final Dimensions

Length: 15 meters Breadth: 12 meters

Alternative Methods for Solving the Same Problem

There are several alternative ways to solve this problem. Here are a few examples:

Using a Direct Equation

If we set the length as X and the breadth as X - 3, we can solve the equation:

X(X - 3) 180

This simplifies to:

X^2 - 3X - 180 0

Solving Using the Quadratic Formula

X (3 √729) / 2 (3 27) / 2 15 (the negative solution is discarded)

Breadth X - 3 12

Using the Square Root Method

The solution can also be found by taking the square root of 180:

√180 13.4164

The sides must be slightly smaller and larger:

15 × 12 180

Therefore, the dimensions are 15 meters × 12 meters.

Conclusion

By following various methods, we can effectively solve for the dimensions of a rectangle given its area and the relationship between its length and breadth. The correct dimensions for the given problem are: