Arectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. with 1000 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

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