The perimeter of a paved area being built at a playground is going to be 46 meters. If the length and width are each a whole number of meters, what should the length and width of the area be to cover the greatest possible area?
let the length be x
and the width be y
2x + 2y = 46
x+y = 23
y = 23 - x
area = xy
= x(23-x) = -x^2 + 23x
vertex of this parabola :
the x of the vertex is -b/(2a) = -23/-2 = 11.5
then y = 23-11.5 = 11.5
So the paved area should be a square with sides 11.5 m each , resulting in a maximum area of 132.25 m^2
(Of course , the largest rectangle for a given perimeter is always a square)