Question

A farmer has 200 m of fencing with which he wishes to enclose a rectangular field.One side of the field can make use of a fence that already exists. What is the maximum area he can enclose?

Answer 1

`5000m^2` is the required area. I have used elementary concepts of maxima and minima.

Explanation:

Let area be `A` and the sides of rectangular field be `x and y;`
So,
`A=x*y`

Now, one side of the rectangle is already made with a fence.
There are 4 sides, two sides of `x` meters and two sides of `y` meters. Let a side of `y` meters be already fenced.

Then, the remaining three sides are to be fenced with a fence of `200m` length,

So,

`2x+y=200`

`y=200-2x`
So,

`A=x(200-2x)`

`A=200x-2x^2`

For area to be maximum,

`(dA)/dx=0`

And,

`(d^2A)/dx^2<0`

Now,

`(dA)/dx=200-4x`

`200-4x=0`

`x=50`

Now,

`(d^2A)/dx^2=-4`

Which is negative

Thus, `x=50` is a maximum.

Now,

`2x+y=200`

`2(50)+y=200`

`y=100`

Thus, the required area is:

`A=x*y=50*100=5000 m^2`