Find length and width of plot?

Question

Farmer Ed has 800 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, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Answer 1

The perimeter will be 2w + l (since you don't have to fence the side next to the river).

So, you can set 2w + l = 800

The area will be w*l which you want to maximize.

Rewrite the first equation so it is set to just one variable on one side.

l=800-2w

Plug it into the expression for area

w*(800-2w)

and then distribute so it's a quadratic expression:

800w-2w² or -2w²+800w

Now how do you find the maximum of that parabola? It's going to be the vertex.

You can use -b/2a for the x (or w) value, and then run it back through the expression to find the y value which will be your maximum area.