Optimization Problems: can you at least help me get started? ?

Draw the semi-circle and the rectangle inside of it. Draw a line from the center of the diameter-line of the semi-circle to the corner of the rectangle. That is R. Define that angle as "a". Put the sides of the rectangle in terms of "a" and "R". Use trigonometry. Define the area of the rectangle in terms of "a" & "R", take the derivative of that in respect to "a" and set it equal to 0. Solve that equation for "a" to find the value of "a" at the local maximum of your definition of the area. Go back and plug that into the definition of the sides of the rectangle. That should leave your area equation in terms of just "R".