Respuesta :
Answer:
[tex]A= 125*250=31250 ft^2[/tex]
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:
[tex]P=2w +l[/tex]
Since they are using the creek as one side.
So then we have this:
[tex]500 =2w +l[/tex] (2)
Now we can solve w in terms of l from eqaution (2) and we got:
[tex]w=\frac{500-l}{2}[/tex] (3)
And we can replace this condition into equation (1) like this:
[tex]A= \frac{500-l}{2} l =250l - \frac{1}{2} l^2[/tex]
And we can maximize this function derivating respect to l and we got:
[tex]\frac{dA}{dl}= 250 -l=0[/tex]
And then we got that [tex]l=250[/tex]
And if we solve for w from equation (3) we got:
[tex]w=\frac{500-250}{2}=125[/tex]
And then the dimensions would be:
[tex] l =250ft , w=125ft[/tex]
And the area would be:
[tex]A= 125*250=31250 ft^2[/tex]
