A rectangle is inscribed in a semicircle 
of radius 2. If the variable x represents 
half the length of the rectangle, express 
the area of the rectangle as a function of x.

Since x represents half the length of the
rectangle, the length of rectangle = 2x

Let y represent the height of the rectangle. 



Then the Area of the rectangle is

Area = length × width

   A = (2x)y

   A = 2xy

However we must now express y in terms of x.

Draw in a radius (which equals 2) from the 
center of the semicircle to the upper right
corner of the rectangle:



Use the Pythagoren theorem on the right triangle:

x² + y² = 2²

x² + y² = 4

     y² = 4 - x²
           ______  
      y = Ö4 - x²

So substitute this for y in

      A = 2xy
             ______
      A = 2xÖ4 - x²

Edwin