Question

Someone please explain this to me.
-A farmer has 400 feet of fencing. He wants to fence off a rectangular field that borders a straight river (no fence along the river). What are the dimensions that will give him the largest area?

Answer 1

This is an optimization problem with a constraint. The farmer has 400 feet of fencing. That means that the total perimeter of the field is going to be 400 feet; or, letting l = length and w = width, 2L+2w=400. From that, we can solve for either L or w; in this case, we have w = 200 - L. What do we do with this? Well, we want to optimize area; area = Lw. Plugging in our substitution, A = L(200-L) = 200L-L^2. To optimize this with respect to L, we take the derivative with respect to L and set it equal to zero - 200 - 2L = 0 => L = 100. Thus a fence with a length of 100 feet will produce the optimal area. We can determine optimal width by plugging this into the perimeter equation, to get that the ideal dimensions are 100x100.

Answer 2

but why does it turn into 200?

Answer 3

Why does what turn into 200?

Answer 4

oh just kidding i got it i got confused for a second thank you though!