Calculate Rectangular Plot Perimeter Using Algebraic Expressions

This problem requires us to find the perimeter of a rectangular plot given its length and breadth as algebraic expressions. We need to first find a general formula for the perimeter in terms of $x$ and $y$, and then substitute the given values of $x=1$ and $y=-1$ to find the numerical perimeter.

Step 1: Understand the Perimeter Formula

The perimeter of a rectangle is calculated using the formula:

$$ P = 2 \times (\text{Length} + \text{Breadth}) $$

We are given:

• Length ($L$) = $ (x^2 + xy + y^2) $ m
• Breadth ($B$) = $ (x^2 - 5xy - y^2) $ m

Step 2: Substitute Expressions into the Perimeter Formula

Now, substitute the given expressions for length and breadth into the perimeter formula:

$$ P = 2 \times \left[ (x^2 + xy + y^2) + (x^2 - 5xy - y^2) \right] $$

Step 3: Simplify the Expression for Perimeter

Combine the like terms inside the brackets:

• Combine $x^2$ terms: $ x^2 + x^2 = 2x^2 $
• Combine $xy$ terms: $ xy - 5xy = -4xy $
• Combine $y^2$ terms: $ y^2 - y^2 = 0 $

So the expression inside the brackets simplifies to:

$$ 2x^2 - 4xy $$

Now, multiply this by 2:

$$ P = 2 \times (2x^2 - 4xy) $$

$$ P = 4x^2 - 8xy $$

The perimeter of the rectangular plot is $ (4x^2 - 8xy) $ meters.

Step 4: Evaluate the Perimeter with Given Values

We are given $x = 1$ and $y = -1$. Substitute these values into the simplified perimeter expression:

$$ P = 4(1)^2 - 8(1)(-1) $$

Calculate the terms:

• $ 4(1)^2 = 4(1) = 4 $
• $ -8(1)(-1) = -8(-1) = 8 $

Add the results:

$$ P = 4 + 8 $$

$$ P = 12 $$

Therefore, the perimeter of the rectangular plot is 12 meters.

Final Answer Verification

Let's check the dimensions with the given values:

• Length = $ (1)^2 + (1)(-1) + (-1)^2 = 1 - 1 + 1 = 1 $ m
• Breadth = $ (1)^2 - 5(1)(-1) - (-1)^2 = 1 - 5(-1) - 1 = 1 + 5 - 1 = 5 $ m
• Perimeter = $ 2 \times (1 + 5) = 2 \times 6 = 12 $ m

The calculated perimeter matches the result obtained using the simplified algebraic expression.