Question

please help ...A farmer wants to enclose a rectangular field along a river on three sides. If 1,200 feet of fencing is to be used, what dimensions will maximize the enclosed area

Answer 1

ok before we start, let me tell you the answer and tell you how i know it without doing any calculus
the answer is that side opposite the river should be 600 (half of your total) and the other two sides should therfore be 300

Answer 2

it always works this way when one side is free. (agains a river, barn, highway, whatever)
if you have to use all 4 sides, then the answer is obviously a square. now we can do the calculus

Answer 3

your total lenght for the three sides is \(2x+y\) so we know \(2x+y=1200\) and thefore \(y=1200-2x\)

Answer 4

area is \(A=xy\) replace \(y\) by \(1200-2x\) to get
\[A(x)=x(1200-2x)=1200x-2x^2\] now this is a parabola that opens down, and so the maximum is at the vertex. vertex is
\[-\frac{b}{2a}=-\frac{1200}{2\times (-2)}=300\] as promised

Answer 5

you can replace 1200 by P for perimeter and check that you will get \(\frac{P}{4}\) for the short side and therefore \(\frac{P}{2}\) for ht elong side

Answer 6

if you have questions let me know, in particular if you need to do this using the derivative we can do it that way too

Answer 7

no you did more than help you actually explained it ! thank you

Answer 8

yw