Answer:
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
