2. A farmer is planning to create a temporary ‘pen’ for livestock, by cutting a large section of plastic netting into strips, joining the strips together and then fixing the combined length to posts. He decides the fence must be 2 m high, and must form a shape that is nearly square: either a precise square or a rectangle whose length and width do not differ by more than 5 m.

If the length and width are both whole numbers of metres, what is the greatest area than can be created from 156 m2 of netting?

The area of the plastic netting is 156 m² and its height is told to be 2 m, so solving the equation of the area for the length, you can calcualte how many linear meters of plastic netting the farmer will use:

Area = length × height

156 m² = length × 2 m ⇒ length = 156 m² / 2m = 78 m

So, you have to create the greatest possible area with 78 linear meters of plastic netting.

The greatest possible area enclosed by a fixed amount perimeter is a square.

If the figure were a precise square each side would be 1/4 of 78m, i.e. 78 m / 4 = 19.5 m.

But you need to use whole numbers that do not differ by more than 5 meters.

Suppose two sides measure 19 m and the other two sides measure 20 m. You can check that makes the perimeter equal to 78 m:

19 m + 19m + 20 m + 20m = 78 m.

That figure is a rectangle whose area is 19 m × 20 m = 380 m²

Since, that is the figure most similar to a square using wholde numbers you know that is the greatest area.

You can prove that using other numbers:

18m + 18m + 21m + 21m = 78m

18m × 21m = 378m²

As you see, the are for a rectanle 18 × 21 is smaller than the area for a rectangle 19 × 20. And the area will decrease as you move further away from the square shape.

Thus, the greatest area is 380m², using two strips 19 meters long and two strips 20 meters long, which is nearly a square.