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Optimisation and best use of space

  1. Mar 24, 2012 #1
    2iv02ub.jpg

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

    I initially had to use the attached diagram to solve problems related to a concert venue.So I created a formula for perimeter and area and used these to create a formula for area with x as the only variable.I used differentiation to find the value of x when the area is at a maximum.

    Value for Pi given as 22/7

    P=4xy + 2x +Pix

    A=4xy +Pix^2/2

    y=p-36/7x

    y into area formula to give a= px -25x^2/7

    The final question i'm stuck on asks "Critically examine the shape of the concert venue with this maximum area and comment on it.Make 2 suggestions for improving the concert venue given that the perimeter must remain the same"

    2. Relevant equations

    I've solved these to

    a =-25x^2/7 + px and x =7p/50


    3. The attempt at a solution

    I'm a little stuck on what the final question is asking, my intial observations are that a circle would give the maximum area based off a fixed perimeter and that while a square offers more area than a rectangle, now it is used in conjunction with a semi-circle the rectangle provides the best solution.As the circle is linked to the x value, if you reduce x to make a square the area of the circle is reduced.


    Thanks for reading
     
    Last edited: Mar 24, 2012
  2. jcsd
  3. Mar 24, 2012 #2

    Mark44

    Staff: Mentor

    Your diagram is so small that I can't read the dimensions on it. Please provide the exact problem statement, including the dimensions of the concert venue.
     
  4. Mar 24, 2012 #3
    Made the necessary changes :)
     
  5. Mar 25, 2012 #4
    So i'm thinking of covering the below in my obeservation:

    How and circle or square are usually the best use of perimeter for maximum area with proven examples in the form of equations.

    How in this example this doesn't work as the circle perimeter is linked proportionally to the x value making a rectangle a better solution

    For the improvements:

    A circular venue would provide the maximum area.

    If I thought in 3D terms, would a tiered seating plan increase the area?


    Is there anything else you think I should add?

    Thanks
     
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