Consider a window that is in the shape of a rectangle topped by a semicircle. Given that the perimeter of the window is 14 meters, express the window's area, A, as a function of w, the width of the window. Note that this type of window is called a Norman window and the width of the rectangle, w, is the diameter of the described semicircle.A = ____ m2

Question

Consider a window that is in the shape of a rectangle topped by a semicircle. Given that the perimeter of the window is 14 meters, express the window&#039;s area, A, as a function of w, the width of the window. Note that this type of window is called a Norman window and the width of the rectangle, w, is the diameter of the described semicircle.A = ____ m2

Brennan Parks · Accepted Answer

(1)My goal is to help you understand this problem.Please let me know if you have any questions about myanswer.(2)The width and height of the rectangle are w and h,respectively.The area and perimeter of the window are A and P,respectively.The perimeter of the rectangular part of the window Is      P    r    =  w  +  2  hThe perimeter of the semicircular part of the window is      P    s    =      1    2    (  2  π  (      w    2    )  )  =      π    2    wTherefore   P  =      P    r    +      P    s    =  (  1  +      π    2    )  w  +  2  hRewriteh in terms of w:  (  1  +      π    2    )  w  +  2  h  =  P  ⇒  2  h  =  P  −  (  1  +      π    2    )  w  ⇒    h  =      P    2    −      1    2    (  1  +      π    2    )  w  ⇒  h  =      P    2    −  (      1    2    +      π        4    )  w(3)The area of the rectangular part of the Window is      A    r   = wh.The area ot the semicircular part ot the windoW Is      A    s    =      1    2    (  π  (            w      2        2    )  =      π    8        w    2  Therefore   A  =      A    r    +      A    s    =      π    8        w    2    +  w  hSubstitute      P    2    −  (      1    2    +      π    4    )  w   for   h  :  A  =      π    8        w    2    +  w  [      P    2    −  (      1    2    +      π    4    w  ]  =  (      π    8    −      1    2    −      π    4    )      w    2    +      P    2    w   A  =  −  (      π    8    +      1    2    )      w    2    +      P    2    w    P  =  14  ⇒  A  =  −  (      π    8    +      1    2    )      w    2    +  7  w(4)Have you already rated another answer?Remember, you can change your rating if you prefer myanswer.

Answered question

Answer & Explanation

New Questions in High school geometry