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Function problem

  1. Sep 16, 2007 #1
    1. The problem statement, all variables and given/known data


    A norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.
    2. Relevant equations

    2l +2w=30

    3. The attempt at a solution
    If this were simply a rectangle,

    I can solve for l in terms of w.

    l=15-w

    A=w(15-w)


    but I don't know how to handle the semicircle part.
    1. The problem statement, all variables and given/known data



    2. Relevant equations

    (pir^2)/2


    3. The attempt at a solution
     
    Last edited: Sep 16, 2007
  2. jcsd
  3. Sep 16, 2007 #2
    First, 2l+2w is NOT 30 ft. The perimeter includes the curved (semi-circle) part, so the perimeter is the sum of the lengths of the two sides, the base (the width 'x'), and the semi-circle.
     
  4. Sep 16, 2007 #3
    So how do I convert that into function form?
     
  5. Sep 16, 2007 #4
    Just add the quantities I stated.

    Length = l, width = x, perimeter of semi-circle = ?
     
  6. Sep 16, 2007 #5
    But the perimeter of the entire window is 30 ft. The length and width of the rectangle/semicircle are unknown.
     
  7. Sep 16, 2007 #6
    OK, here's what I came up with.

    A(w)=w(30-w-pir^2)/2
     
  8. Sep 16, 2007 #7

    symbolipoint

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    You are asked to find a function; not to compute a value. A value of width or length is not needed.

    r=radius, L=length, p=pi
    area= L*2*r + 0.5*p*r^2;
    perimeter= 2*r + 2*L + p*r = 30

    You only have two variables there. Any found value of r will determine the corresponding value for L (the other dimension of the rectangle).. There is enough information to find the function of area based on ONE variable, the radius, r.
    I used 2*r as the width, so you may want to rewrite some of the above shown equations based on "x" instead of 2*r.
     
  9. Sep 16, 2007 #8
    So in solving for r, I get (30-2l-ps)/2

    where ps is perimeter of semicircle.

    But I want that in r in terms of l right?
     
  10. Sep 16, 2007 #9
    putting the equation in terms of width, I got w(30-r-Ps)

    P being perimeter of the semicircle.

    How does that look?
     
    Last edited: Sep 16, 2007
  11. Sep 16, 2007 #10

    symbolipoint

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    Maybe changing the variables made the information confusing.
    Let us try:
    x=width of rectange, same as diameter of the semicircle;
    L=length of rectangle
    p= Pi=close to 3.1415....
    A= area

    A= 0.5*p*(x/2)^2 + x*L
    and
    30 = p*(x/2) + 2*L + x

    You want A in terms of x. Use the second equation and substitute expression for L into the first equation.
     
  12. Sep 16, 2007 #11
    relate the length or the arc of the semi circle to the variable of the rectangle (w and or l)

    the eqation of a circle's cicomference is C=2pi*r correct? you soul not need any more information
     

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    Last edited: Sep 16, 2007
  13. Sep 16, 2007 #12
    Now I am completely lost. The diameter would be c/pi? The perimeter would be pir-2r+2r+2l. or pir+2l?
     
    Last edited: Sep 16, 2007
  14. Sep 16, 2007 #13
    there is a picture attached to my post, use the picture

    if you can't see it i will explain it.

    i denoted the bottom of the window as W
    the sides as L
    and the arc perimiter= unknown

    however we know the radius of the arc is just W/2
    so r=W/2

    the formula of the perimiter of a circle is C=2pi*r
    so given that R=W/2 what is the circumference?
    *hint for next step...this is not a circle it is a semi circle*
    what is the cirumference of the semi circle
     
  15. Sep 16, 2007 #14
    attachment pending approval. You are saying use the circumference minus diameter plus diameter (the base) plus the two lengths for perimeter?
     
  16. Sep 16, 2007 #15
    Ok see what you think of this.

    A(w)= w((30-w-pir)/2)+(3-w-2H)/pi


    w=width or diameter

    H=height

    r=radius
     
    Last edited: Sep 16, 2007
  17. Sep 16, 2007 #16
    not exactly sure what you are saying. I might have worded my question incorrectly.

    i meant the circumference of the semi circle (not including the base...only the arc)
    which is essentialy half of the circumference of the circle.
    wha did you get fo the circumference of the circle?


     
  18. Sep 16, 2007 #17
    i appologize humbly i may have misundertsood the question.

    so i read the questionover again and now it has confused me....its a rectangle with a semi circle up top right, and you want to find the area of the circle (as an equation) but your given that 2l+2w=30 (or did you do that?) because the 2L i understand, but shouldnt it be 2L+w+perimiter of semi circle.

    and the area is esentialy the area of rectangle +area of semi circle
    and the area of the rectangle is L*W
    and the area of the semi circle is _____________ (what we are trying to figure out)
    so the formula for the area of a circle is pi*r^2
    and r=(w/2)
    so area of semi circle is pi(W/2)^2


    so that means A=L*W+pi(W/2)^2


    but they want it in terms of x (a single variable...however you have L and w in there which is 2 variables...so how would you make that one variable?)
    *hint* has somethign to do with whats given....that the perimitr is 30....which is 2L+w+the perimiter of the arc of the semi circle....
    and so what i was doing earlier was trying to get you to create an equation that relates the L and W to the perimiter of the semi circle
     
    Last edited: Sep 16, 2007
  19. Sep 16, 2007 #18
    All I was given is the perimeter of the norman window, which is a recatangle surmounted by a semicircle, is equal to 30. Express the area of the window as a function of the width of the window. Sorry, the description I gave originally is straight from the calculus text. I got pir for the circumference of the semicircle.
     
  20. Sep 17, 2007 #19

    HallsofIvy

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    Exactly. The window has one base, of length x, two heights of length y, and a semicircle of length [itex]\pi r[/itex]. Have you drawn a picture? Can you see what r is in terms of x? You know that the total length is 30. What equation is that?

    Now, what is the area of an x by y rectangle? What is the area of the semi circle?
     
  21. Sep 17, 2007 #20
    H=(30-pi(w/2)-w)/2

    A(w)=w(30-pi(w/2)-w)/2+.5pi(w/2)^2
     
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