Question

you have 276 feet of fencing to enclose a rectangular region. what is the maximum area that?

Answer 1

here is a handy fact. The maximum occurs when it's a square

Answer 2

Let x & y be the sides of the rectangle.
perimeter 2x+2y=276
x+y=138
y=138-x
area A=x*y=x(138-x)=138x-x^2
\[\frac{ dA }{ dx }=138-2x,\frac{ dA }{ dx }=0 ~gives138-2x=0,x=69\]
\[at~x=69,\frac{ d^2A }{ dx^2 }= -2 ~<0,\]
there is a maxima at x=69
Hence A is maximum when x =69
find y