A window is being built and the bottom is a rectangle and the top is a semicircle. If the perimeter is 12 meters, what dimensions of the window would let in the most light

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      

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