Answer:
length  =  45 units
height  =  19,84 units
A(max) =  892,18 units²
 Â
Step-by-step explanation: Â Â ( See annex)
r = 30 units
Area of rectangle is:    A  = 2*p*h   (by symmetry)
First, we have to get p and h as function of x
Look at triangle AOC Â and see:
p = r - x     and   h = √ (30)²  - p²      or   h = √(30)²  - (30 - x )²
we can simplify the expresion and get
h  =  √(30)²  -  [  (30)² +  x² - 60*x ]
h  =  √60*x - x²   for simplicity reason let say  [60*x - x² ] = z
Then we have
A(x)  = 2 * p * h  ⇒ A(x)  = 2 * ( 30 - x ) * √(60*x - x²  Â
A(x) = (60 - 2x ) √(60x - x²
A(x)  = 60√(60x - x²)   -  2x √(60x - x²)
We are rady to take derivative
A´(x)  =  0 +[ 60* 1/2 * ( 60 - 2x ) ] / √(60x - x²)  - 2 √(60x - x²) -
      2x *1/2 *( 60 - 2x ) ] / √(60x - x²)
Developing such expresion
A´(x)  =  [ 1800 - 60x / √(60x - x²) - 2√(60x - x² -  [60x  -2x²] /√(60x -x²
A´(x)  = { [ 1800 - 60x ]  - 2 (60x - x² ) -  60x - 2x² } /√(60x -x²
Then  A´(x) = 0
        [ 1800 - 60x ]  - 2 (60x - x² ) -  60x - 2x²  = 0
1800 - 240 *x  = 0
 240* x  = 1800     x = 1800/240
x = 7.5 units  and   p  = r - x   ⇒  p = 30 -7,5 =  p = 22,5
and  h = √60*(7,5) - (7,5)²
h = 19,84 Â units
A(max) = 2* 22,5 * 19,84 Â
A(max) = 892, 8 un²   we can compare this figure with the area of semicircle (1413 un²) and with areas of squares close in dimensions
lets say  square of side 23   which is 529 un² Â
