Enclosing the MOst Area with a fence

[karliekay]

A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed.


it states that the width is x and the length is 4000-2x.

I know that the Perimeter is 4000. So, 4000=L+2w because we are not using on of the lengths.

What I did is inputed 4000-2x in for L, but I got 0. What do I do?

 

[masters]

A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed.


it states that the width is x and the length is 4000-2x.

I know that the Perimeter is 4000. So, 4000=L+2w because we are not using on of the lengths.

What I did is inputed 4000-2x in for L, but I got 0. What do I do?

Hi karliekay,

You have a rectangle with one side being the river.

We know the width (the sides perpendicular to the river) to be $\displaystyle x$ meters.
The length (the side parallel to the river) is $\displaystyle 4000-2x$

Area = width X length

$\displaystyle A(x) = x(4000-2x)$

The graph of this quadratic function is a parabola. Where the graph crosses the x-axis, the value of the area (A) must be zero. Therefore, our equation becomes:

$\displaystyle x(4000-2x)=0$

So our graph crosses the x-axis when $\displaystyle x = 0$ (on its way up) and again when $\displaystyle (4000 - 2x) = 0$ (on its way down). Solving the second factor, we get:

$\displaystyle 4000-2x=0$

$\displaystyle -2x=-4000$

$\displaystyle x=2000$

This tells you that the graph crosses the x-axis at 0 and 2000. Since a parabola is symmetrical, the x-coordinate of the vertex will be halfway between these two values: $\displaystyle \frac{0+2000}{2}=1000$ meters.

So the maximum area occurs when

The width = $\displaystyle x = 1000$ meters.

The length = $\displaystyle 4000-2x = 4000-2 \cdot 1000= 2000$ meters.

The maximum area is $\displaystyle 2000 \cdot 1000 = 2000000$ square meters.