A rectangle is constructed with its base on the diameter of a semicircle with radius 30 and with its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum​ area?

Answer:length  =  45 unitsheight  =  19,84 unitsA(max) =  892,18 units²    Step-by-step explanation:     ( See annex)r = 30 unitsArea of rectangle is:      A  = 2*p*h    (by symmetry)First, we have to get p and h as function of xLook at triangle AOC  and see:p = r - x         and     h = √ (30)²  - p²           or     h = √(30)²  - (30 - x )²we 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 ] = zThen we haveA(x)  = 2 * p * h  ⇒ A(x)  = 2 * ( 30 - x ) * √(60*x - x²    A(x) = (60 - 2x ) √(60x - x²A(x)  = 60√(60x - x²)    -  2x √(60x - x²)We are rady to take derivativeA´(x)  =  0 +[ 60* 1/2 * ( 60 - 2x ) ] / √(60x - x²)  - 2 √(60x - x²) -             2x *1/2 *( 60 - 2x ) ] / √(60x - x²)Developing such expresionA´(x)  =  [ 1800 - 60x / √(60x - x²) - 2√(60x - x² -  [60x  -2x²] /√(60x -x²A´(x)  = { [ 1800 - 60x ]  - 2 (60x - x² ) -  60x - 2x² } /√(60x -x²Then  A´(x) = 0                 [ 1800 - 60x ]  - 2 (60x - x² ) -  60x - 2x²  = 01800 - 240 *x  = 0  240* x  = 1800         x = 1800/240x = 7.5 units   and    p  = r - x    ⇒  p = 30 -7,5 =   p = 22,5and  h = √60*(7,5) - (7,5)²h = 19,84  unitsA(max) = 2* 22,5 * 19,84  A(max) = 892, 8 un²    we can compare this figure with the area of semicircle (1413 un²) and with areas of squares close in dimensionslets say  square of side 23    which is 529 un²