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Calculus Word + Linear Problem (2)

  1. Sep 19, 2013 #1
    hello I am new here and just starting to do some maths in school (not always my strong point) and I am need of help for some problems that I am trying to solve!

    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 37 feet, express the area of the window as a function of the width (across the base) of the window.

    a(x) = ?

    2. Relevant equations

    some equations used are x*y = area or 2x + 2y and πr^2

    3. The attempt at a solution

    from my calculations I thought the answers would be 37x-x^2, but apparently it isn;t, and I dont have any solutions to confirm my answer(s). THis probably seems easy to you guys, but word problems have me clueless.

    1. The problem statement, all variables and given/known data

    THis second question is a linear matrix algebra type of question...

    Solve the system:

    1 1 0 0 : -1
    0 1 1 0 : 3
    0 0 1 1 : -1
    1 0 0 1 : -5

    Now normally i know how to solve this but this time its asking for the solution in the form of this and dont know how to input the values:

    [x1] = [] []
    [x2] = [] + [] s.
    [x3] = [] []
    [x4] = [] []


    2. Relevant equations

    Regular algebra...but haven't seen this type of question yet so there is perhaps something i dont know?

    3. The attempt at a solution

    I attempted solving it such as a normal matrix and then solving for x1,x2 etc...and after inputting the numbers my answers was incorrect.

    i had x1 = -1 + -1 x2 = 3 + -1 x3 = -1 + -1 x4 = 1 -4
     
    Last edited: Sep 19, 2013
  2. jcsd
  3. Sep 19, 2013 #2

    tiny-tim

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    hello sheet1! welcome to pf! :smile:
    what is your formula for L (the length of the rectangle) as a function of W (the width?)
    i don't understand what this means :confused:

    (the solution looks like a constant vector V that would be a solution to a "zero" equation, plus a parameter s times a particular solution vector W: V + sW)
     
  4. Sep 19, 2013 #3

    SteamKing

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    First, you've got to figure how the perimeter of the window relates to ts dimensions.
     
  5. Sep 19, 2013 #4
    Ok I edited the first one and added a(x)= ? ... thats all the information I have on that question, and for the second one its hard to describe since I can't add a picture:

    its formatted as such:

    x1 + x2 = -1
    x2 + x3 = 3
    x3 + x4 = -1
    x1 + x4 = -5

    then there is 3 groups in a square bracket, one with x1 2,3,4 in a vertical line equaling another square box with 4 vertical blank solutions following by that plus (+) another 4 vertical boxes, the question is very similar to this youtube video, i followed it but could not get a correct answer...:

     
    Last edited by a moderator: Sep 25, 2014
  6. Sep 19, 2013 #5

    Ray Vickson

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    So, what steps have you taken in solving the system? You are required to actually _show_ what you did.
     
    Last edited by a moderator: Sep 25, 2014
  7. Sep 19, 2013 #6
    for the first question I did something like this (I have lots of notes scribbles, many attempts, easier to say what I did rather than write stuff out here, since it is much harder):

    perim=x+2*h+Pi*x=37 => h=(37-x-Pi*x)/2
    A(x)=h*x+Pi*x^2=(37-x-Pi*x)*x+Pi*x^2=37x-x*x

    for the systems I just isolated the values and solved like normally, but what I DONT understand is what they are asking for in the question, the video i showed above shows it in detail of how to solve it, but after following it step by step I keep getting a wrong answer and am unaware on how to solve a system described in such a way...i need help! i just want to learn..
     
  8. Sep 19, 2013 #7

    tiny-tim

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    ah, nooo :redface:

    perim = x + 2*h + Pi*x/2 :wink:
     
  9. Sep 19, 2013 #8
    ok thanks for that, i got the solution! -.8926990813x^2 + 18.5x ... now i really need help with the linear problem, please with a cherry on top with chocolate whipped cream.
     
  10. Sep 19, 2013 #9

    tiny-tim

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    ok, show us how far you got :smile:
     
  11. Sep 19, 2013 #10
    ok i got this far :

    a = -1 -b
    b = 3 -c
    c = -1 -d
    d = -5 -a

    a being x1 b being x2 and so forth...just not sure what to do from this point on. If I plug one of the values into another they just cancel out each other, was I supposed to use gauss-Jordan elimination even further?
     
  12. Sep 19, 2013 #11
    I can tell you've worked out how to get the answer but your 10th significant figure is wrong!
     
  13. Sep 19, 2013 #12
    not according to the computer it aint! oay, do u think you could shed some light on my linear equation?
     
  14. Sep 20, 2013 #13

    SteamKing

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    I don't think the question asks you to find the numerical value of the width, just express the area A of the window in terms of the width W, subject to the constraint that the perimeter of the window is 37 feet.

    Since the window is a rectangle with a semi-circle on top, one can see that the diameter of the semi-circle D must be equal to the width W of the rectangle below.

    The perimeter of the semi-circle is
    π * D / 2, or π * W / 2
    The perimeter of the rectangle below is
    W + 2 * H

    The total perimeter of the window is P =
    π * W / 2 + W + 2 * H
    = 37 feet

    P =
    (1 + π/2)*W + 2*H
    = 37 feet

    Solve for H.

    Then, calculate the area of the semi-circle and the rectangle and add them together.

    I have no idea why you wrote all those linear equations to solve.
     
  15. Sep 20, 2013 #14

    Ray Vickson

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    He said it was a second question. He just mis-formatted his posting.
     
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