Solving a Rectangular Area Problem Using Quadratic Equations

Introduction

In this article, we will explore a problem related to the dimensions of a rectangle and how to solve it using a quadratic equation. The problem involves finding the breadth of a rectangle given the relationship between its length and breadth, and its area. This type of problem is a common application of quadratic equations in real-world scenarios, such as in geometry and engineering.

Problem Statement

The length of a rectangle is 3 meters greater than its breadth. If the area is 70 square meters, what is its breadth?

Step-by-Step Solution

Let's denote the breadth of the rectangle by b meters. Consequently, the length, l, can be expressed as:

l b 3

The area (A) of a rectangle is given by the formula:

A l × b

Given that the area is 70 square meters, we can write:

70 (b 3) × b

A. Expansion and Simplification

Expanding the equation, we get:

70 b^2 3b

Subtract 70 from both sides to form a standard quadratic equation:

b^2 3b - 70 0

B. Solving the Quadratic Equation

Using the quadratic formula to solve for b, where the quadratic equation is in the form ax^2 bx c 0 and the quadratic formula is:

x frac{-B ± √{B^2 - 4AC}}{2A}

Here, A 1, B 3, and C -70. Plugging in these values, we find the discriminant:

3^2 - 4(1)(-70) 9 280 289

The square root of 289 is 17. Hence, the solutions for b are:

b frac{-3 ± 17}{2}

Calculating the two possible solutions:

b frac{-3 17}{2} frac{14}{2} 7

b frac{-3 - 17}{2} frac{-20}{2} -10

Since the breadth cannot be negative, we discard -10 and accept 7 as the valid breadth.

Therefore, the breadth of the rectangle is 7 meters.

Alternative Solutions and Verification

Alternatively, we can factorize the quadratic equation:

b^2 3b - 70 0

Factorizing by finding two numbers that multiply to give -70 and add up to 3, we find that the numbers are 10 and -7.

Hence, the equation can be rewritten as:

(b 10)(b - 7) 0

Setting each factor to 0 and solving for b gives us:

b 10 0 or b - 7 0

b -10 or b 7

Again, we discard -10 and accept 7 as the valid breadth.

Geometric Interpretation and Practical Application

The problem demonstrates how quadratic equations can be used to solve real-world geometric problems. In this case, we found that a rectangle with a breadth of 7 meters and a length of 10 meters (since length breadth 3) has an area of 70 square meters.

This solution is useful in various fields, including architecture, engineering, and construction, where precise measurements are critical.

Conclusion

Solving problems involving the dimensions of a rectangle using quadratic equations is a valuable skill in mathematics. The detailed steps provided in this article should help you understand the process and apply similar techniques to other similar problems.