Expressing Diagonal Length Of A Restaurant Napkin In Terms Of Its Length
In the realm of geometry and practical applications, even the simplest objects can hold mathematical intrigue. Take, for instance, a rectangular napkin gracing the tables of a restaurant. This seemingly ordinary item becomes a fascinating subject when we delve into its dimensions and their relationships. Let's unravel the mathematical puzzle presented by a restaurant's rectangular napkins, where the length (l) is twice the width, and the diagonal length is denoted by x. Our goal is to express x in terms of l, unveiling the connection between these dimensions.
Setting the Stage: The Rectangular Napkin
Imagine a crisp, white rectangular napkin laid out neatly on a restaurant table. This napkin, like any rectangle, possesses two key dimensions: length and width. In our scenario, the length (l) takes center stage, being twice as long as the width. This establishes a fundamental relationship between the napkin's sides, setting the foundation for our mathematical exploration. Let's denote the width of the napkin as w. According to the problem statement, the length l is twice the width w, which we can express mathematically as:
l = 2w
This equation forms the cornerstone of our analysis, allowing us to connect the length and width in a precise manner. Furthermore, the problem introduces the diagonal of the napkin, denoted by x. The diagonal stretches across the rectangle from one corner to the opposite corner, creating a line segment that cuts through the napkin's interior. Our ultimate objective is to express the length of this diagonal (x) in terms of the length (l). To achieve this, we need to invoke the Pythagorean theorem, a fundamental principle in geometry that governs the relationship between the sides of a right-angled triangle.
The Pythagorean Theorem: Our Guiding Light
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem provides a powerful tool for relating the sides of a rectangle, as the diagonal divides the rectangle into two right-angled triangles. In our case, the diagonal (x) acts as the hypotenuse, while the length (l) and width (w) form the other two sides of the right-angled triangle. Applying the Pythagorean theorem, we can write:
x² = l² + w²
This equation represents a crucial link between the diagonal (x), length (l), and width (w) of the napkin. However, our goal is to express x solely in terms of l. To achieve this, we need to eliminate w from the equation. Fortunately, we have already established the relationship between l and w: l = 2w. We can use this equation to express w in terms of l and then substitute it into the Pythagorean equation.
Expressing Width in Terms of Length
From the equation l = 2w, we can easily express the width (w) in terms of the length (l) by dividing both sides of the equation by 2:
w = l / 2
Now we have a direct relationship between w and l, allowing us to substitute this expression into the Pythagorean equation. This substitution will eliminate w and leave us with an equation involving only x and l.
The Substitution: Unveiling the Connection
Substituting w = l / 2 into the Pythagorean equation x² = l² + w², we get:
x² = l² + (l / 2)²
This equation now involves only x and l, bringing us closer to our goal of expressing x in terms of l. To simplify the equation further, we need to expand the squared term and combine like terms.
Expanding the squared term, we have:
x² = l² + l² / 4
To combine the terms, we need a common denominator. We can rewrite l² as 4l² / 4, giving us:
x² = 4l² / 4 + l² / 4
Now we can combine the terms:
x² = 5l² / 4
This equation represents a significant step forward, as it directly relates the square of the diagonal (x²) to the square of the length (l²). To find x in terms of l, we need to take the square root of both sides of the equation.
Taking the Square Root: The Final Step
Taking the square root of both sides of the equation x² = 5l² / 4, we get:
x = √(5l² / 4)
We can simplify this expression further by using the properties of square roots. The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator:
x = √5 * √(l²) / √4
The square root of l² is l, and the square root of 4 is 2, so we have:
x = (√5 * l) / 2
This is our final expression for x in terms of l. It reveals that the length of the diagonal (x) is equal to the square root of 5 times the length (l), all divided by 2.
The Grand Finale: Expressing x in Terms of l
Therefore, the length of the diagonal (x) of the rectangular napkin, in terms of its length (l), is:
x = (√5 / 2) * l
This elegant equation encapsulates the mathematical relationship between the diagonal and the length of the napkin. It demonstrates how the Pythagorean theorem, combined with careful algebraic manipulation, allows us to solve geometric problems and express dimensions in terms of one another. This seemingly simple napkin has unveiled a fascinating mathematical connection, showcasing the power of geometry in everyday objects.
In conclusion, by applying the principles of geometry, particularly the Pythagorean theorem, we have successfully expressed the diagonal (x) of the rectangular napkin in terms of its length (l). The final result, x = (√5 / 2) * l, highlights the interconnectedness of mathematical concepts and their relevance in understanding the world around us. This exercise not only provides a solution to a specific problem but also reinforces the importance of mathematical reasoning and problem-solving skills in various contexts.
