Question

Farmer Ed has 7000 meters of fencing and wants to enclose a rectangular plot that borders on a river. If farmer Ed does not fence the side along the river, what is the largest are that can be enclosed? ...?

Answer 1

Let

x--------> the length side of the rectangular plot (assume side along the river)

y------> the width side of the rectangular plot

we know that

the perimeter of the rectangular plot is equal to

`P=x+2y`

`P=7,000\ m`

so

`7,000=x+2y`

Clear variable y

`y=(7,000-x)/2` -------> equation
`1`

The area of the rectangular plot is

`A=x*y` ------> equation
`2`

substitute equation
`1` in equation
`2`

`A=x*[(7,000-x)/2]`

`A=x*[(7,000-x)/2]\\\\ A=3,500x-0.5x^(2)`

we know that

To find the larger area that can be enclosed------> Find the vertex of the quadratic equation

The quadratic equation is a vertical parabola open down

so

the vertex is a maximum

using a graphing tool

see the attached figure

the vertex is the point
`(3,500,6,125,000)`

that means

For
`x=3,500\ m`

the largest area is
`6,125,000\ m^(2)`

Find the value of y

`y=(7,000-3,500)/2=1,750\ m`

the dimensions of the rectangular plot are

`Length=3,500\ m\\Width=1,750\ m`

therefore

the answer is

The largest area that can be enclosed is
`6,125,000\ m^(2)`