Respuesta :
Answer:
x  =  0,41 m   or   x  =  41 cm
r  =  2,18 m
Step-by-step 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   Â
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*Ï€ + 16 ) - 24*Ï€ + 48 Â = 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
