Answer :
To find the perimeter of the garden, we need to first identify and solve for the value of [tex]x[/tex] using the given area of the garden.
**Formula for Area: **
The formula for the area of a rectangle is given by:
[tex]\text{Area} = \text{Length} \times \text{Breadth}[/tex]**Substitute the Given Values: **
We have [tex]\text{Length} = (x + 10)[/tex] meters and [tex]\text{Breadth} = (x + 6)[/tex] meters. Given the area is 396 square meters, we can set up the equation:
[tex](x + 10)(x + 6) = 396[/tex]**Expand and Simplify the Equation: **
Expanding the left side, we get:
[tex]x^2 + 6x + 10x + 60 = 396[/tex]
Simplifying this, we have:
[tex]x^2 + 16x + 60 = 396[/tex]**Rearrange into a Quadratic Equation: **
Bring all terms to one side to form a quadratic equation:
[tex]x^2 + 16x + 60 - 396 = 0[/tex]
[tex]x^2 + 16x - 336 = 0[/tex]**Solve the Quadratic Equation: **
We can solve this using the quadratic formula:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
where [tex]a = 1[/tex], [tex]b = 16[/tex], and [tex]c = -336[/tex].**Calculate the Discriminant: **
[tex]b^2 - 4ac = 16^2 - 4(1)(-336)[/tex]
[tex]256 + 1344 = 1600[/tex]**Find the Roots: **
[tex]x = \frac{-16 \pm \sqrt{1600}}{2}[/tex]
[tex]x = \frac{-16 \pm 40}{2}[/tex]
Solving these gives:
[tex]x = \frac{24}{2} = 12[/tex] or [tex]x = \frac{-56}{2} = -28[/tex]Since [tex]x[/tex] must be a positive value (as lengths cannot be negative), [tex]x = 12[/tex].
**Find the Length and Breadth: **
Plug [tex]x = 12[/tex] back into the expressions for length and breadth:- Length: [tex]x + 10 = 12 + 10 = 22[/tex] meters
- Breadth: [tex]x + 6 = 12 + 6 = 18[/tex] meters
**Calculate the Perimeter: **
The perimeter [tex]P[/tex] of a rectangle is given by:
[tex]P = 2(\text{Length} + \text{Breadth})[/tex]
[tex]P = 2(22 + 18) = 2 \times 40 = 80[/tex] meters
Therefore, the perimeter of the garden is [tex]80[/tex] meters.
