# Optimization Using Differentiation

izmeh
[SOLVED] Optimization Using Differentiation

I have an assignment in which we are to optimize problems using a given 6-step process. More or less it involves Max/Min differentiation.

On of the problems are as follow;
Enclosing the Largest Area
The owner of the Rancho Los Feliz has 3000 yd of fencing material to enclose the rectangular piece of grazing land along the striaght portion of a river. If fencing is not required along the river, what are the dimensions of the lagrgest area that the he can enclose? What is the area?

I under stand that...
a=xy
p=2x+2y

i understand one of the sides can be added to the other 3 sides, however, i'm not sure how to make this a function.

Have you learned how to use Lagrange multipliers to handle constraints?

NateTG
Homework Helper

Try figuring out the area of the enclosure based on the lenght of one of the sides:
A(x)=blah blah blah.

you should be able to construct a function like that.

P.S. The farmer doesn't have to put fence where the river is, so you should probably use:
p=2x+y

ShawnD
What you have to do is substitute one into the other.
a = xy
p = x + 2y

lets rearrange the perimeter equation in terms of x
x = p - 2y
and since we know p, we can can a bit farther
x = 3000 - 2y

since x = 3000 - 2y
a = (3000 - 2y)y
a = 3000y - 2y^2

since the area changes when we change the y, lets find when our differential
da/dy = 3000 - 4y
since the maximum area is when the area stops increasing, we equate to 0
0 = 3000 - 4y
after solving, we get
y = 750

now fill that back into our original equation of x = 3000 - 2y
x = 3000 - 2(750)
now solve
x = 1500

now fill in for the area formula, a = xy
a = 1500 * 750
a = 1125000

Right on 