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.

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