Maximizing total area of a pen?

Question

Build a rectangular pen with three parallel partitions using 500 feet of fencing. What deminsions will maximize the total area of the pen?

Answer 1

OK... lemme draw a diagram:

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if I'm understanding this correctly, the fencing will have to build five sides of length x, and two sides of length ((500-5x)/2).

The total area of the pen will be x*((500-5x)/2).
That's -2.5x^2 + 250x
This is the equation of a downward-opening parabola. We're interested in finding its tip, or vertex, since that's where area is maximized.

The equation of the parabola factors into:
-2.5x * (x - 100)
Obviously, this equals zero when x=0 or x=100.
Since parabolas are symmetrical, this means the vertex is where x=50.

If x=50, then the other side is length (500-5*50)/2, or or 250/2, or 125.

The pen has dimensions 50*125 feet.

Answer 2

usually in this case a square maximizes the area. I don't understand the "three parallel partitions" part, so ... anyways, build a square using the 500 feet as the perimeter.
If you wanna know why, go to www.artofproblemsolving.com and ask someone there.

Answer 3

83 , 84 are the required dimensions.

as 4* 83 + 2 * 84 =500,
satisfies the given conditions.
also as their product 83*84 has to be the maximum as they are consecutive nos.

Answer 4

era in-between answer: Very exciting. looks to me, 3 achievable shapes: circle, rectangular (3 squares), triangle (equilateral). rectangular looks to be wasteful of fencing for the section completed. i will sleep on it.