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7 February, 04:54

Tanya has 16 yards of fencing to enclose a rectangular garden. Consider the possible dimensions of the rectangular garden with a perimeter of 16 yards. Determine, in square yards, the largest area that can be contained by this amount of fence.

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  1. 7 February, 05:03
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    The largest area can be contained by this amount of fence is 16 square yards

    Step-by-step explanation:

    The formula of the perimeter of a rectangle is P = 2l + 2w, where l is its length and w is its width The formula of the area of a rectangle is A = l * w

    Assume that the length of the rectangular garden is x yards and its width is y yards

    ∵ The length of the fence is 16 yards

    ∴ The perimeter of the garden is 16 yards

    ∵ P = 2l + 2w

    ∵ The length of the garden is x yards

    ∵ The width of the garden is y yards

    ∴ P = 2x + 2y

    - Equate the formula of P by 16

    ∴ 2x + 2y = 16

    - Divide both sides by 2 to simplify the equation

    ∴ x + y = 8

    - Find y in terms of x

    - Subtract x from both sides

    ∴ y = 8 - x ⇒ (1)

    ∵ A = l * w

    ∴ A = x * y

    - Substitute y by equation (1)

    ∴ A = x (8 - x)

    - Simplify the right hand side

    ∴ A = 8x - x²

    To find the largest are differentiate A with respect to x and equate the differentiation by 0 to find the value of x which gives the maximum dimensions to get the largest area

    ∵ A' = 8 - 2x

    ∵ A' = 0

    ∴ 0 = 8 - 2x

    - Add 2x to both sides

    ∴ 2x = 8

    - Divide both sides by 2

    ∴ x = 4

    Substitute the value of x in equation of the area

    ∵ A = 8x - x²

    ∴ A = 8 (4) - (4) ²

    ∴ A = 32 - 16

    ∴ A = 16 yards²

    The largest area can be contained by this amount of fence is 16 square yards
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