Question

A farmer wants to fence a rectangular pasture using a river as one side. The pasture must contain 180,000 square meters of space for the herd to frolic. What is the least amount of fencing required to enclose such a pasture?

Answer 1

1200 m

Explanation:

We are given that

Area of pasture,A=180000 square meter

We have to find the least amount of fencing required to enclose such a pasture.

Let length of pasture,=x

Width of pasture,b=y

Area of pasture=
`l* b=xy`

`xy=180000`

`y=(180000)/(x)`

Fencing required=2x+y

`P=2x+(180000)/(x)`

Differentiate w.r.t x

`(dP)/(dx)=2-(180000)/(x^2)`

`(dP)/(dx)=0`

`2-(180000)/(x^2)=0`

`(180000)/(x^2)=2`

`x^2=(180000)/(2)=90000`

`x=√(900000)=300`m

It is always positive because length cannot be negative.

`y=(180000)/(300)=600`m

Again differentiate w.r.t x

`(d^2P)/(dx^2)=(360000)/(x^3)`

Substitute x=300

`(d^2P)/(dx^2)=(360000)/(27000000)=0.013>0`

The fence is minimum at x=300

Therefore, fencing required to enclose such a pasture

`P=2x+y=2(300)+600=1200 m`