High School

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------------------------------------------------ The length and breadth of a garden are (x + 10) metres and (x + 6) metres respectively.

Find the perimeter of the garden if the area of the garden is 396 square metres.

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.

  1. **Formula for Area: **
    The formula for the area of a rectangle is given by:
    [tex]\text{Area} = \text{Length} \times \text{Breadth}[/tex]

  2. **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]

  3. **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]

  4. **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]

  5. **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].

  6. **Calculate the Discriminant: **
    [tex]b^2 - 4ac = 16^2 - 4(1)(-336)[/tex]
    [tex]256 + 1344 = 1600[/tex]

  7. **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].

  8. **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
  9. **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.