1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
natta225 [31]
2 years ago
14

A window is being built and the bottom is a rectangle and the top is a semicircle. If the perimeter is 12 meters, what dimension

s of the window would let in the most light
Mathematics
1 answer:
EleoNora [17]2 years ago
4 0

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      

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

You might be interested in
Solve the equation using inverse operations. Check your solutions. In your final answer, include all of your work.
natali 33 [55]
X = − 3 + 3+3i√3
————————
2
,3−3i√3
—————
2
5 0
3 years ago
Read 2 more answers
THIS TOO GOD I NEED HELP SO BAD I DONT UNDERSTAND
Zina [86]

Answer:

54.7

Step-by-step explanation:

16.7 + 11 + 11 + 8 + 8 = 54.7

Note

I don't know how you are wrong but I showed my work and I got the same answer which is weird.

8 0
2 years ago
What is it p l z <br><br>zzzzzzzzzzzzzzzzz
Aleks04 [339]
Since a whole rotation is 360 degrees, you can take the whole rotation subtract it by the minor arc (92 degrees) which will equal the major arc (B=268 degrees)
5 0
10 months ago
Read 2 more answers
Witch expression is equivalent to 6^-3
Arturiano [62]

Do you have any options of the equivalent expressions or is it just a plan question that you just asked?



4 0
3 years ago
Read 2 more answers
POSSIBLE PO
Tcecarenko [31]

Answer:

707 inches squared

Step-by-step explanation:

Find the area of each side of the rectangular prism then add all of the areas together.

hope this helps

have a good day

4 0
1 year ago
Other questions:
  • I need to know asap plz
    14·2 answers
  • Which are zeros of the function represented by this graph?
    7·1 answer
  • Maureen makes $15 an hour at her job, plus an 18% bonus on her salary for that day if she sells a super item. Maureen worked 5 h
    6·1 answer
  • Write the equation of a line with a slope of 3 and a y-intercept of 4
    7·1 answer
  • one possible combination of homeroom groups is 18 groups of 10 students and 5 groups of 12 students. which equation uses the x =
    12·1 answer
  • Nathan is playing golf at a golf course. The height of the golf ball from the ground, in feet, x seconds after the ball is hit i
    11·1 answer
  • Suppose a random variable, x, arises from a binomial experiment. If n = 25, and p = 0.85, find the variance. Round answer to 4 d
    6·1 answer
  • True or False?
    7·1 answer
  • Reflect each point in the x-axis and y-axis
    14·2 answers
  • Find the area of the shaded region.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!