Finding The Width Of A Rectangle With Area 91 Square Inches

In the fascinating world of geometry, rectangles hold a special place. Their simple yet elegant form makes them fundamental building blocks in various applications, from architecture to engineering. Understanding the relationships between their dimensions – length, width, and area – is crucial for solving practical problems and gaining deeper insights into mathematical concepts. In this article, we embark on a journey to explore a specific rectangular puzzle. We are given the area of a rectangle, which is 91 square inches, and a peculiar relationship between its length and width: the length is 1 inch less than twice its width. Our mission is to write an equation that will help us find the elusive width of this rectangle.

Before we dive into the specifics of our problem, let's revisit the fundamental concept of the area of a rectangle. The area of a rectangle is the space it occupies within its boundaries, and it's calculated by multiplying its length (l) by its width (w). Mathematically, this is expressed as:

Area = length × width or A = l × w

This simple formula forms the cornerstone of our investigation. It connects the two dimensions of the rectangle to its area, providing a vital link for solving our puzzle. Now that we've refreshed our understanding of the area of a rectangle, let's delve into the unique characteristics of our specific rectangular challenge.

Our rectangle isn't just any ordinary rectangle; it possesses a distinctive feature. The length of our rectangle is intricately linked to its width. We are told that the length is 1 inch less than twice its width. This seemingly simple statement holds the key to unraveling the rectangle's dimensions. Let's translate this verbal description into a mathematical expression. If we represent the width of the rectangle by the variable 'w', then twice the width would be 2w. Since the length is 1 inch less than twice the width, we can express the length (l) as:

l = 2w - 1

This equation beautifully captures the relationship between the length and width of our rectangle. It allows us to express the length in terms of the width, which will be crucial in formulating our final equation. Now that we have both the area formula and the relationship between length and width, we can combine them to create the equation we seek.

We know that the area of the rectangle is 91 square inches, and we have an expression for the length in terms of the width. It's time to put these pieces together and construct the equation that will lead us to the width of the rectangle. We start with the area formula:

A = l × w

We know that A = 91 and l = 2w - 1. Substituting these values into the area formula, we get:

91 = (2w - 1) × w

This equation is a powerful statement that connects the known area with the unknown width. It's a quadratic equation, which means it involves a variable raised to the power of 2. To make it easier to solve, let's expand the equation and rearrange it into the standard quadratic form:

91 = 2w^2 - w

Subtracting 91 from both sides, we get:

2w^2 - w - 91 = 0

This is the equation we've been striving for! It's a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = -1, and c = -91. This equation encapsulates the essence of our rectangular puzzle. Solving this equation will reveal the possible values for the width (w) of the rectangle.

Now that we have our equation, 2w^2 - w - 91 = 0, the next step is to solve it for 'w'. There are several methods for solving quadratic equations, including factoring, using the quadratic formula, or completing the square. For this particular equation, let's use the quadratic formula, which is a versatile tool that works for any quadratic equation.

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 2, b = -1, and c = -91. Plugging these values into the quadratic formula, we get:

w = (1 ± √((-1)^2 - 4 × 2 × (-91))) / (2 × 2)

Simplifying the expression under the square root:

w = (1 ± √(1 + 728)) / 4

w = (1 ± √729) / 4

Since the square root of 729 is 27, we have:

w = (1 ± 27) / 4

This gives us two possible solutions for w:

w₁ = (1 + 27) / 4 = 28 / 4 = 7

w₂ = (1 - 27) / 4 = -26 / 4 = -6.5

We've arrived at two possible solutions for the width of the rectangle: 7 inches and -6.5 inches. However, in the world of geometry, dimensions like width cannot be negative. A negative width doesn't make sense in the physical context of a rectangle. Therefore, we discard the negative solution and accept the positive solution as the valid width of our rectangle.

So, the width of the rectangle is 7 inches. Now that we know the width, we can use the relationship between length and width to find the length:

l = 2w - 1

l = 2 × 7 - 1

l = 14 - 1

l = 13

Therefore, the length of the rectangle is 13 inches.

In this mathematical exploration, we successfully unraveled the dimensions of a rectangle given its area and a unique relationship between its length and width. We started by understanding the fundamental concept of the area of a rectangle and then translated the verbal description of the length-width relationship into a mathematical expression. By combining these elements, we formulated a quadratic equation that encapsulated the essence of our problem.

We then employed the quadratic formula to solve the equation, obtaining two possible solutions for the width. Recognizing that a negative width is not physically meaningful, we discarded the negative solution and accepted the positive solution as the valid width. Finally, we used the width to calculate the length of the rectangle.

This journey through the world of rectangles highlights the power of mathematical equations in solving real-world problems. It demonstrates how algebraic concepts can be applied to geometric scenarios, providing us with a deeper understanding of the shapes and spaces around us. The rectangle, with its simple yet profound properties, continues to be a source of mathematical fascination and a testament to the elegance of geometric relationships.

The key to solving this rectangle problem lies in understanding the relationship between its dimensions and area. Understanding the area of a rectangle is crucial for solving geometric problems. The equation 2w^2 - w - 91 = 0 helps find the width of the rectangle.