Find the dimensions of the shaded region so that its area is maximized. (enter your answers as a comma-separated list.)
sides:
height:

Interesting problem!
The equation of the circle is
f(x)=y=sqrt(9-x^2)
Since the figure is symmetrical about the line y=x, we conclude that the inverse function is also
f(y)=x=sqrt(9-y^2)
 
Let the height of the trapezoid be h, then the x-coordinate of the top side with the circle is given by
f(y)=x=sqrt(9-y^2)=sqrt(9-h^2)

The area of the trapezoid is the
A(h)=h*(6+2*sqrt(9-h^2))/2 with h=independent variable

Differentiate with respect to h, equate to zero to find the maximum/minimum,
A'(h)=(3sqrt(9-h^2)-2h^2+9)/sqrt(9-h^2)=0 
consequently at maximum/minimum,
(3sqrt(9-h^2)-2h^2+9)=0
rearrange and square,
9*(9-h^2)=(2*h^2-9)^2
simplify to 
h^2(4h^2-27)=0
h=0 or h=+/- sqrt(27)/2
reject h=0 (minimum) and negative value of h to give
h=sqrt(27)/2=2.5981

Length of top side
= 2sqrt(9-h^2)
=2sqrt(9-27/4)
=sqrt(36-27)
=3
Length of slanted side
=sqrt(1.5^2+3^2)
=3

Therefore sides from top are {3,3,6,3}
h=sqrt(27)/2