The ride has 100 metres of fencing to make a rectangular enclosure as shown.

link for the image imageshack com
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It will use existing walls for two sides of the enclosure, and leave an opening of 2 metres for a gate.
a Show that the area of the enclosure is given by:
A = 102x – x2
b Find the value of x that will give the maximum possible area.
C Calculate the maximum possible area

To show that the area of the enclosure is given by A = 102x - x^2, we can break down the problem as follows:

1. The total length of the fencing available is 100 meters.
2. Since there are two existing walls, the remaining length of the fence is (100 - 2) = 98 meters that needs to be used for the other two sides.
3. Let's assume the length of one of the sides perpendicular to the existing walls is x meters. The corresponding width will then be (98 - x) meters.

Now, to calculate the area of the enclosure:

Area = Length × Width
= x × (98 - x)
= 98x - x^2

Hence, the area of the enclosure is given by A = 102x - x^2.

To find the value of x that will give the maximum possible area, we can differentiate the area equation A = 102x - x^2 with respect to x and set it equal to zero.

dA/dx = 102 - 2x

Setting dA/dx = 0, we have:

102 - 2x = 0
2x = 102
x = 51

So, the value of x that will give the maximum possible area is 51 meters.

To calculate the maximum possible area, substitute x = 51 into the area equation:

A = 102x - x^2
= 102(51) - (51^2)
= 5202 - 2601
= 2601 square meters

Therefore, the maximum possible area of the enclosure is 2601 square meters.