Hint

Let x be the height for the trapezoid inscribe in a semicircle of radius 10. Then

represents the shorter base of the trapezoid and the area is given by

Note the maximum area occurs when the height is about 8.66 and at that point the length of the shorter base would be or, approximately 10.

Would it make more sense to set up the problem letting x represent the length of the shorter base?

In this case the long base is 20, the short base is x and the height is so the area is

The graph looks like:

It appears that the maximum area occurs when the short base is the same as the radius of the semicircle.