Question

2. A farmer has 10000m fencing wire with much to fence three sides of his rectangular farm,the fourth side being existing fence of his neighbour.find in metres the dimension of the field of the largest possible area that can be enclosed ?

Answer 1

`2500` by `5000` metres.

Explanation:

We call the length `l` and the width `w`. Therefore,

`2L + W = 10000`

Area is given by

`A = LW`

If we solve for `W` in the first equation, we get:

`W = 10000 - 2L`

Substituting into the second, we get:

`A = L(10000 - 2L)`

`A = 10000L - 2L^2`

If we differentiate with respect to `L`, we get:

`A' = 10000 - 4L`

Maximums and minimums occur when the derivative equals `0`.

`0 = 10000 - 4L`

`4L = 10000`

`L = 2500`

If we want to check if it's a maximum, we may choose to select test points. At `x = 0`, the derivative is positive, that's to say the function is increasing. At `x = 3000`, the derivative is negative, that 's to say the function is decreasing.

Therefore, `L= 2500` is indeed a maximum.

This means that the width is

`2(2500) + W = 10000 -> W = 5000" meters"`

Hopefully this helps!