Question 87300
<pre>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.
<font size = 4 color = "indigo"><b>
Since x represents half the length of the
rectangle, the length of rectangle = 2x

Let y represent the height of the rectangle. 

{{{drawing(400,196,-4,4,0,4,circle(0,0,2.1), line(-2,0,2,0),
rectangle(-1.29,0,1.29,1.6),locate(-.1,.18,o),
locate(-.7,.3,x), locate(.7,.3,x), locate(1.3,.9,y), locate(-.2,1.6,"2×") 

)}}}

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:

{{{drawing(400,196,-4,4,0,4,circle(0,0,2.1), line(-2,0,2,0),
rectangle(-1.29,0,1.29,1.6),locate(-.1,.18,o),  line(0,0,1.3,1.6),
locate(-.7,.3,x), locate(.7,.3,x), locate(1.3,.9,y), locate(-.2,1.6,"2×"), 
locate(.4,1,"2")
)}}}

Use the Pythagoren theorem on the right triangle:

x² + y² = 2²

x² + y² = 4

     y² = 4 - x²
           ______  
      y = <font face = "symbol">Ö</font>4 - x²

So substitute this for y in

      A = 2xy
             ______
      A = 2x<font face = "symbol">Ö</font>4 - x²

Edwin</pre>