We need a formula for area as a function of the height (y)

parabola: y = 27 - x^{2}

So: the width of the rectangl is simply 2x

and th height is y.

We cn write a formula for the area:

A(x ) = y*2x = (27-x^{2})* 2x = 54x - 2 x^{3}

To find th maximum area, find the first derivative an set = to 0:

A' = 54 -6x^{2} = 0

x^{2 }=^{ }9 and so x = +/-3

finally, A = 54*3 - 2*27 =108

CHECK by plotting using a graphing utility.