Answer:
length = 45 units
height = 19,84 units
A(max) = 892,18 units²
Step-by-step explanation: ( 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 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
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 units
A(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 dimensions
lets say square of side 23 which is 529 un²
