Opimization problem?

Question

A farmer has 160 feet of fence and wants a pen to adjoin to the whole side of the 130 foot barn as shown.

http://i144.photobucket.com/albums/r162/patel748/3.png

What should the dimentions be for maximum area. Note that x is greater then or equal to 0 but less then or equal to15.

Yeah, I got the same answer as you. I am pretty much stumped on how to get an answer in between 0 and 15 for x.

Answer 1

The amount of fencing needed is

2x + y + (y - 130) = 160
2x + 2y = 290
y = (290 - 2x) / 2 = 145 - x

Also,

y >= 130
x <= 15

The area, which is to be maximized, is

A = x * y
= x * (145 - x)
= 145x - x^2

If we merely take the first derivative of this and set it equal to zero, we will have no way to incorporate the two constraints on the values of x and y. So let us take the first derivative and look at it:

f'(x) = 145 - 2x

This is the rate at which the area is changing for any given value for x. At x = 0, the area is increasing at a rate of 145 for each unit increase in x. When x = 1, the area is increasing at a rate of 143 for each unit increase in x. You can see that the area continues to increase even as x gets up to its maximum of 15.

The answer is:
x = 15
y = 130
A = 1950 (maximum)

There is no value for x between 0 and 15 that yields the maximum area. Only x = 15 yields the maximum area.

Answer 2

and why does the farmer what to know the dimentions ???
makes no sense
there has to b a reason