Answer:
The largest area is 125000 m²
The dimensions of the farmland are 250 m and 500 m
Step-by-step explanation:
* Lets pick the information from the problem
- The farmland is shaped a rectangle
- The farmland will be bounded on one side by a river
- The other three sides are bounded by a single-strand electric fence
- The length of wire is 1000 m
- Lets consider the width of the rectangle is x and the length is y
- The side which will be bounded by the river is y
∴ The perimeter of the farmland which will be bounded by the electric
fence = x + x + y = 2x + y
- We will use the wire to fence the farmland
∵ The length of the wire is 1000 m
∵ The perimeter of the farmland is equal to the length of the wire
∴ 2x + y = 1000
- Lets find y in term of x
∵ 2x + y = 1000 ⇒ subtract 2x from both sides
∴ y = 1000 - 2x
- Now lets find the area can enclose by the wire
∵ The area of the rectangle = length × width
∵ The width of the farmland is x and its length is y
∴ The area of the farmland (A) = x × y = xy ⇒ (2)
- Use equation (1) to substitute the value of y in equation (2)
∴ A = x (1000 - 2x) ⇒ simplify
∴ A = 1000 x - 2 x²
- To find the maximum area we will differentiate A with respect to x
and equate the answer by zero to find the value of x which will make
the enclosed area largest
* Lets revise the rule of differentiation
- If y = ax^n, then dy/dx = a(n) x^(n-1)
- If y = ax, then dy/dx = a
- If y = a, then dy/dx = 0 , where a is a constant
∵ A = 1000 x - 2 x² ⇒ (3)
- Differentiate A with respect to x using the rules above
∴ dA/dx = 1000 - 2(2) x^(2-1)
∴ dA/dx = 1000 - 4x
- Put dA/dx = 0 to find the value of x
∵ 1000 - 4x = 0 ⇒ add 4x to both sides
∴ 1000 = 4x ⇒ divide both sides by 4
∴ 250 = x
∴ The value of x is 250
- Lets substitute this value in equation 3 to find the largest area
∵ A = 1000 x - 2 x²
∴ A = 1000 (250) - 2(250)² = 125000 m²
* The largest area is 125000 m²
∵ The width of the farmland is x
∵ x = 250
∴ The width of the farmland = 250 m
- Substitute the value of x in the equation (1) to find y
∵ y = 1000 - 2x
∵ x = 250
∴ y = 1000 - 2(250) = 1000 - 500 = 500
∵ The length of the farm lend is y
∴ The length of the farm land = 500 m
* The dimensions of the farmland are 250 m and 500 m
