Finding The Length Of A Rectangle Using Area And Width
In the realm of geometry, rectangles stand as fundamental shapes, characterized by their four sides and four right angles. Understanding the relationships between a rectangle's dimensions – its length, width, and area – is crucial in various mathematical and real-world applications. This article delves into the intricacies of determining a rectangle's length when its area and width are known, providing a step-by-step guide to the process and highlighting key concepts.
Defining the Rectangle: Area, Length, and Width
Before embarking on the calculation, let's solidify our understanding of the core components of a rectangle:
- Area: The area of a rectangle represents the two-dimensional space it occupies, typically measured in square units (e.g., square inches, square centimeters). It is calculated by multiplying the rectangle's length and width.
- Length: The length of a rectangle refers to its longer side.
- Width: The width of a rectangle refers to its shorter side.
The relationship between these elements is encapsulated in the following formula:
Area = Length × Width
This formula serves as the cornerstone for solving problems involving rectangles, allowing us to determine any one of the three variables if the other two are known.
The Challenge: Finding Length from Area and Width
Our primary objective is to determine the length of a rectangle when its area and width are provided. This task necessitates rearranging the area formula to isolate the length variable. Let's explore the algebraic manipulation involved:
Starting with the formula:
Area = Length × Width
To isolate Length, we divide both sides of the equation by Width:
Length = Area / Width
This derived formula provides the key to calculating the length of a rectangle, given its area and width. The area, which represents the total space within the rectangle, and the width, the shorter side measurement, are the crucial pieces of information. By dividing the area by the width, we effectively reverse the multiplication process, revealing the length – the longer side measurement. This concept is fundamental in geometry and has practical applications in various fields, from construction and design to everyday problem-solving.
A Practical Example: Unveiling the Rectangle's Length
Let's apply our newfound knowledge to a concrete example. Consider a rectangle with an area of $45x^2 - 42x - 48$ and a width of $5x - 8$. Our mission is to determine the length of this rectangle.
Following the formula we derived, we have:
Length = (45x^2 - 42x - 48) / (5x - 8)
This expression presents a polynomial division problem. To solve it, we can employ either long division or synthetic division. For the sake of clarity, we will demonstrate the long division method here.
Long Division: A Step-by-Step Approach
Long division, a familiar arithmetic technique, extends seamlessly to polynomial expressions. Let's embark on the process:
- Set up the division: Write the dividend ($45x^2 - 42x - 48$) inside the division symbol and the divisor ($5x - 8$) outside.
____________________
5x - 8 | 45x^2 - 42x - 48
- Divide the leading terms: Divide the leading term of the dividend ($45x^2$) by the leading term of the divisor ($5x$). The result ($9x$) becomes the first term of the quotient.
9x_________________
5x - 8 | 45x^2 - 42x - 48
- Multiply and subtract: Multiply the entire divisor ($5x - 8$) by the first term of the quotient ($9x$) and write the result ($45x^2 - 72x$) below the dividend. Subtract this result from the dividend.
9x_________________
5x - 8 | 45x^2 - 42x - 48
- (45x^2 - 72x)
____________________
30x - 48
- Bring down the next term: Bring down the next term of the dividend (-48) to form the new dividend (30x - 48).
9x_________________
5x - 8 | 45x^2 - 42x - 48
- (45x^2 - 72x)
____________________
30x - 48
- Repeat the process: Divide the leading term of the new dividend ($30x$) by the leading term of the divisor ($5x$). The result (6) becomes the next term of the quotient.
9x + 6____________
5x - 8 | 45x^2 - 42x - 48
- (45x^2 - 72x)
____________________
30x - 48
- Multiply and subtract again: Multiply the entire divisor ($5x - 8$) by the new term of the quotient (6) and write the result ($30x - 48$) below the new dividend. Subtract this result from the new dividend.
9x + 6____________
5x - 8 | 45x^2 - 42x - 48
- (45x^2 - 72x)
____________________
30x - 48
- (30x - 48)
____________________
0
- The quotient is the answer: The quotient obtained from the long division is $9x + 6$. Therefore, the length of the rectangle is $9x + 6$.
Factoring: An Alternative Approach
Factoring the quadratic expression $45x^2 - 42x - 48$ can simplify the division process. First, we look for a common factor among the coefficients. In this case, 3 is a common factor, so we can rewrite the expression as $3(15x^2 - 14x - 16)$. Next, we factor the quadratic expression inside the parentheses. This involves finding two binomials that, when multiplied, yield $15x^2 - 14x - 16$. After some trial and error, or by using a systematic approach like the AC method, we find that the quadratic factors into $(3x - 8)(5x + 2)$. So, the area expression becomes $3(3x - 8)(5x + 2)$. Now, we can express the length as:
Length = [3(3x - 8)(5x + 2)] / (5x - 8)
Notice that we have a factor of $(5x - 8)$ in both the numerator and the denominator. Canceling these common factors, we are left with:
Length = 3(3x + 2) = 9x + 6
This matches the result we obtained using long division, reinforcing the accuracy of both methods. Factoring can sometimes be a quicker route, especially when the expressions involved have easily identifiable factors. However, long division is a more general method that works even when factoring is not straightforward.
Identifying the Dividend: The Role of the Area
In the division process, the area expression, $45x^2 - 42x - 48$, plays the crucial role of the dividend. The dividend is the quantity that is being divided, and in this context, it represents the total area of the rectangle. It's the starting point for our division, the value we're breaking down into the product of the length and the width. Understanding this role is key to setting up the division problem correctly and interpreting the results.
Conclusion: Mastering Rectangle Dimensions
In this comprehensive exploration, we've unveiled the method for determining the length of a rectangle when its area and width are known. By rearranging the fundamental area formula and employing polynomial division techniques, we successfully navigated a practical example. Whether through long division or factoring, the underlying principle remains consistent: dividing the area by the width reveals the length. This understanding empowers us to tackle a wide array of geometric problems and appreciate the interconnectedness of mathematical concepts.
This understanding of rectangles' dimensions extends beyond textbook exercises. It is a foundational concept in various practical applications, such as architectural design, where calculating room dimensions and material requirements is essential, and in landscaping, where determining the area of a garden or patio helps in planning layouts and purchasing materials. Even in everyday situations, such as arranging furniture in a room or estimating the size of a rug, the principles of length, width, and area come into play. Therefore, mastering these concepts not only enhances mathematical skills but also provides valuable tools for problem-solving in real-world scenarios.
By grasping the relationship between area, length, and width, and by practicing the techniques of polynomial division and factoring, we equip ourselves with a powerful toolkit for geometric analysis and problem-solving. The ability to manipulate these concepts allows us to approach mathematical challenges with confidence and apply our knowledge to practical situations with ease. This journey through the dimensions of rectangles underscores the elegance and utility of mathematical principles in our world.
