Solving For Dimensions Of An Enlarged Rectangle A Mathematical Exploration
Let's dive into a fascinating geometrical problem that involves rectangles, areas, and a touch of algebraic manipulation. Picture this: we've got an original rectangle with a length we'll call "C" and a width labeled "L". Now, we decide to bump up the size of this rectangle by adding the same amount, "X", to both its length and width. The puzzle is this if the area of this new, enlarged rectangle is a whopping 7 times the area of the original rectangle, what exactly are the dimensions of this new rectangle?
Setting Up the Equations: The Foundation of Our Solution
To crack this mathematical enigma, we need to translate the problem's words into the language of equations. This is where the magic of algebra comes into play. Let's start by expressing the areas of both rectangles.
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Original Rectangle's Area: The area of a rectangle, as we all know, is simply its length multiplied by its width. So, the area of our original rectangle is C * L. We can write this as:
Area_original = C * L -
New Rectangle's Area: We've increased both the length and width by "X". This means the new length is C + X, and the new width is L + X. The area of the new rectangle is then:
Area_new = (C + X) * (L + X)
Now, here's the crucial piece of information the problem gives us the new rectangle's area is 7 times the original rectangle's area. This translates directly into the equation:
Area_new = 7 * Area_original
Substituting our expressions for the areas, we get:
(C + X) * (L + X) = 7 * (C * L)
This equation is the key to unlocking the dimensions of the new rectangle. It tells us the relationship between the original dimensions (C and L), the increase (X), and the factor by which the area has grown (7). Our next step is to unravel this equation and see what it reveals.
Expanding and Simplifying: Taming the Algebraic Beast
Our equation, (C + X) * (L + X) = 7 * (C * L), is a bit like a tangled knot. To understand it better, we need to expand and simplify it. This involves multiplying out the terms and rearranging them to get a clearer view of the relationships between C, L, and X.
Let's start by expanding the left side of the equation, (C + X) * (L + X). Using the distributive property (or the FOIL method, if you prefer), we get:
(C + X) * (L + X) = C * L + C * X + X * L + X * X
Which we can write more neatly as:
CL + CX + LX + X^2
Now, let's bring down the right side of the equation, which is simply 7 * (C * L), or 7CL. So, our equation now looks like this:
CL + CX + LX + X^2 = 7CL
We're making progress! To further simplify, let's move all the terms to one side of the equation. We'll subtract 7CL from both sides, giving us:
CL + CX + LX + X^2 - 7CL = 0
Combining the CL terms, we get:
X^2 + CX + LX - 6CL = 0
This equation is a quadratic equation in disguise. It might not look like your typical quadratic equation (ax^2 + bx + c = 0), but it shares the same fundamental structure. Our next challenge is to figure out how to solve this equation for X. This will involve some clever algebraic techniques and a bit of insight.
Solving for X: Unearthing the Hidden Increase
Our simplified equation, X^2 + CX + LX - 6CL = 0, holds the key to finding the value of X, the amount by which we increased the sides of the rectangle. However, it's not a straightforward quadratic equation in the usual form because it contains two variables, C and L, in addition to X. This means we can't directly apply the quadratic formula.
Instead, we need to think of this equation as a quadratic in X, where the coefficients involve C and L. To make this clearer, let's rewrite the equation slightly:
X^2 + (C + L)X - 6CL = 0
Now we can see it more clearly as a quadratic equation in the form aX^2 + bX + c = 0, where:
- a = 1
- b = (C + L)
- c = -6CL
To solve for X, we can use the quadratic formula:
X = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in our values for a, b, and c, we get:
X = (-(C + L) ± √((C + L)^2 - 4 * 1 * (-6CL))) / (2 * 1)
Let's simplify this expression step by step. First, let's focus on the part under the square root:
(C + L)^2 - 4 * 1 * (-6CL) = C^2 + 2CL + L^2 + 24CL
Combining the CL terms, we get:
C^2 + 26CL + L^2
Now, our expression for X looks like this:
X = (-(C + L) ± √(C^2 + 26CL + L^2)) / 2
This is a significant step forward. We've expressed X in terms of C and L. However, the expression under the square root is still a bit complex. Our next challenge is to see if we can simplify it further. Sometimes, these expressions can be factored or manipulated in ways that make them easier to work with.
Simplifying the Square Root: A Quest for Elegance
Our expression for X contains a square root: √(C^2 + 26CL + L^2). Simplifying this square root would make our expression for X much cleaner and easier to interpret. Let's see if we can find a way to do that.
The expression inside the square root, C^2 + 26CL + L^2, looks tantalizingly close to a perfect square trinomial. A perfect square trinomial is an expression that can be factored into the form (a + b)^2 or (a - b)^2. However, our expression doesn't quite fit this pattern. If it were C^2 + 2CL + L^2, it would be simply (C + L)^2. The 26CL term is throwing us off.
Let's try to manipulate the expression to see if we can reveal a hidden pattern. We can rewrite 26CL as 2CL + 24CL. This gives us:
C^2 + 2CL + L^2 + 24CL
The first three terms now form a perfect square trinomial, (C + L)^2. So, we have:
(C + L)^2 + 24CL
Unfortunately, this doesn't lead to a further simplification of the square root. We're stuck with the square root of a sum, and there's no general way to simplify that further without knowing specific values for C and L.
This means our expression for X,
X = (-(C + L) ± √(C^2 + 26CL + L^2)) / 2
is as simplified as it's going to get in general terms. We have expressed the increase X in terms of the original dimensions C and L. Now, let's think about what this means for the dimensions of the new rectangle.
Finding the New Dimensions: The Grand Finale
We've successfully found an expression for X, the amount by which the sides of the original rectangle were increased. Now, we're ready to determine the dimensions of the new rectangle. Remember, the new length is C + X, and the new width is L + X. We'll use our expression for X to find these dimensions.
We have two possible values for X, corresponding to the ± sign in our quadratic formula solution:
X = (-(C + L) + √(C^2 + 26CL + L^2)) / 2
and
X = (-(C + L) - √(C^2 + 26CL + L^2)) / 2
However, since X represents an increase in length, it must be a positive value. The second solution, with the minus sign before the square root, will always be negative (because the square root term will be larger in magnitude than -(C + L)). Therefore, we can discard the negative solution and focus on the positive one:
X = (-(C + L) + √(C^2 + 26CL + L^2)) / 2
Now we can find the dimensions of the new rectangle:
- New Length: C + X = C + (-(C + L) + √(C^2 + 26CL + L^2)) / 2
Simplifying this, we get:
New Length = (C - L + √(C^2 + 26CL + L^2)) / 2
- New Width: L + X = L + (-(C + L) + √(C^2 + 26CL + L^2)) / 2
Simplifying, we get:
New Width = (-C + L + √(C^2 + 26CL + L^2)) / 2
These expressions give us the dimensions of the new rectangle in terms of the original length (C) and width (L). To get numerical values for the dimensions, we would need to know the specific values of C and L. However, we have successfully solved the problem in a general sense, expressing the new dimensions in terms of the original ones.
Conclusion: A Mathematical Journey Complete
We've journeyed through the world of rectangles, areas, and algebraic equations to solve this intriguing problem. We started with a simple scenario an original rectangle being enlarged and ended up with a general solution for the dimensions of the new rectangle. This journey involved:
- Translating the problem into equations.
- Expanding and simplifying algebraic expressions.
- Solving a quadratic equation.
- Manipulating square roots.
- Interpreting the results in the context of the problem.
This problem beautifully illustrates the power of algebra to solve geometrical puzzles. It also highlights the importance of careful simplification and attention to detail in mathematical problem-solving. While we couldn't get specific numerical answers without knowing the original dimensions, we successfully derived general expressions for the new dimensions, showcasing the elegance and power of mathematical reasoning.
