Question

The back of Jill's property is a creek. Jill would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a corral. If there is 500 feet of fencing available, what is the maximum possible area of the corral?

Answer 1

Answer:

$A= 125*250=31250 ft^2$

Step-by-step explanation:

Let's define some notation first :

w= width , l = length , A= Area, P perimeter

For this case we want to maximize the Area given by this function:

A= l w   (1)

With the following restriction P=500 ft

We know that the perimeter on this case is given by:

$P=2w +l$

Since they are using the creek as one side.

So then we have this:

$500 =2w +l$   (2)

Now we can solve w in terms of l from eqaution (2) and we got:

$w=\frac{500-l}{2}$   (3)

And we can replace this condition into equation (1) like this:

$A= \frac{500-l}{2} l =250l - \frac{1}{2} l^2$

And we can maximize this function derivating respect to l and we got:

$\frac{dA}{dl}= 250 -l=0$

And then we got that $l=250$

And if we solve for w from equation (3) we got:

$w=\frac{500-250}{2}=125$  

And then the dimensions would be:

$l =250ft , w=125ft$

And the area would be:

$A= 125*250=31250 ft^2$