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Math Help - Norman window and greatest possible area of a rectangle

  1. #1
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    Question precalculus problems

    Hi everybody I'm a precalculus student, and I'd like some help to understand these two problems?

    1 A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=1-x2. What are the dimensions of such a rectangle with the greatest possible area?

    2-A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 26 feet?

    Thanks in advance!
    Last edited by skorpiox; February 8th 2009 at 02:54 PM.
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  2. #2
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    Hello, skorpiox!

    2) A Norman window has the shape of a semicircle atop a rectangle
    so that the diameter of the semicircle is equal to the width of the rectangle.
    What is the area of the largest possible Norman window with a perimeter of 26 feet?
    Code:
                  * * *
              *           *
            *               *
           *                 *
    
          *                   *
          * - - - - * - - - - *
          |    r         r    |
          |                   |
        h |                   | h
          |                   |
          |                   |
          * - - - - - - - - - *
                   2r

    The circumference of the semicircle is: \pi r
    The perimeter of the rectangle is: 2r + 2h

    The total perimeter is 26 feet.
    . \pi r + 2r + 2h \:=\:26 \quad\Rightarrow\quad h \:=\:13-\frac{\pi+2}{2}r .[1]


    The area of the semicircle is: \tfrac{1}{2}\pi r^2
    The area of the rectangle is: 2rh

    The total area is: . A \;=\;\tfrac{1}{2}\pi r^2 + 2rh .[2]


    Substitute [1] into [2]: . A \;=\;\tfrac{1}{2}\pi r^2 + 2r\bigg[13-\frac{\pi+2}{2}r\bigg]
    . . which simplfies to: . A \;=\;26r - \frac{\pi+4}{2}\,r^2


    And that is the function we must maximize . . .

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  3. #3
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    Hello,

    1 A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=1-x2. What are the dimensions of such a rectangle with the greatest possible area?
    So let a and b the abscissas the base :

    Code:
      |
      |
      |
     _|___,______,____
      |    a      b
      |
    The upper corners points C and D are on the parabola.
    Consider that C has an abscissa of b and D an abscissa of a (since it's a rectangle, with a base on the x-axis, they have the same abscissa)
    b>a

    C has coordinates (b,1-b) and D : (a,1-a) (remember that C and D are on the parabola)
    But since it's a rectangle, CD is parallel to AB, and thus parallel to the x-axis. Hence C and D have the same ordinate
    So in fact D has coordinates (a,1-b)

    Hence 1-b=1-a
    ---> a-b=0 ---> (a-b)(a+b)=0
    So either a=b, which would make an area of 0, either a=-b.
    ---> a=-b.
    Since b is the greatest, b is positive.


    The area is given by AB \times BC
    Distance AB=(b-a)=2b
    Distance BC=1-b

    So A(b)=2b(1-b^2)
    This is the function to maximize.


    Remember that b is between 0 and the positive y-intercept of the parabola (1-x=0 --> x=1) (because the rectangle is inscribed and because b>0)
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  4. #4
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    Quote Originally Posted by skorpiox View Post
    Hi everybody I'm a precalculus student, and I'd like some help to understand these two problems?

    1 A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=1-x2. What are the dimensions of such a rectangle with the greatest possible area?

    2-A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 26 feet?

    Thanks in advance!
    1) A = 2xy = 2x(1-x^2) ... locate the vertex of the parabolic graph representing area to find the maximum.

    2) check this older thread ...

    http://www.mathhelpforum.com/math-he...w-problem.html
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