Answer:
x = 0,41 m or x = 41 cm
r = 2,18 m
Explanation:
Window which allow most light is that with maximum area
The perimeter of the window is the perimeter of the semicircle (psc ) plus the perimeter of the three sides of the rectangle, sides of the rectangle are 2*x and 2*r then:
pt = 12 = psc + pr
psc = π*r pr = 2*x + 2*r
pt = π*r + 2*x + 2*r
pt = 12 = π*r + 2*x + 2*r (1)
The area of the window is:
A(w) = area of the semicircle ( π*r²/2 ) + area of the rectangle (x*2*r)
A(w) = ( π*r²/2 ) + 2*x*r
Using (1) we get: 12 = r* ( π + 2 ) + 2*x
r = ( 12 - 2*x ) / ( π + 2 )
Plugging that value in A(w) we find total area A(w) as a function of x
A(x) = π* [ (12 - 2*x ) / ( π + 2 )]²/ 2 + 2*x*( 12 - 2*x ) / ( π + 2 )
A(x) = π* [ 144 + 4*x² - 48*x/ ( π + 2 )² + 24*x + 4*x²/ ( π + 2 )
A(x) = [ π*/ ( π + 2 )² ] * (144 + 4*x² - 48*x ) + 24*x + 4*x²/ ( π + 2 )
Tacking derivatives on both sides of the equation:
A´(x) = [ π*/ ( π + 2 )² ]* 8*x - 48 + ( 24 + 8*x )/ ( π + 2 )
A´(x) = 0 [ π*/ ( π + 2 )² ]* 8*x - 48 + ( 24 + 8*x )/ ( π + 2 ) = 0
π* (8*x - 48) / ( π + 2 ) + 24 + 8*x = 0
8*π*x - 48*π + ( 24 + 8*x ) *( π + 2 ) = 0
8*π*x - 48*π + 24*π + 48 + 8*π*x + 16*x = 0
16*π*x - 24*π + 16*x + 48 = 0
x ( 16*π + 16 ) - 24*π + 48 = 0
66,24 * x = 75,36 - 48
x = 27,36 / 66,24 m
x = 0,41 m or x = 41 cm
r = ( 12 - 2*x ) / ( π + 2 )
r = 12 - 2* 0,41 / 3,14 + 2
r = 11,18 / 5,14
r = 2,18 m