A farmer wants to fence an area of 6 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

Question

anonymous · Answer

you have to use 2 equations. Since its a rectangle and you know the area u can say the length is y and the width is x. Therefore the equation for the area is xy=6(10^6). 
Next you know that the amount of fence you are gonna use is equal to the perimeter and the fence that splits it into 2. so P=2(x+y) + x

from the first equation solve for y: y=6(10^6)/x and input into 2nd equation:

P=3x+12(10^6)/x

Differentiate:
P'=3-12(10^6)/(x^2)
After a little rearrangement 
P'=(3x^2-12(10^6))/(x^2)
which yields the points 
x=-2000,0,2000
reject the negative and 0 value leaving x to b 2000.
Solve for y in the first equation and you'll get that y =3000