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

ok so the answer to your question is simple as long as you look at the different dimensions that contain the exemplary letters that combined in the right order give you words that eventually within the 5th dimension will conform and give you the correct answer to your question. hope i , stop looking at this easily, look at it with a spectral view.

step-by-step explanation:

told you

3

step-by-step explanation:

divide 13.5 by 9 to get the scale factor. it equals 1.5

2 * 1.5 = 3