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Optimization expression Problem

  1. May 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Hello,

    Can you help me to understand the question..Ineed clarification and hints to solve the question..

    A circle and a square are to be constructed from a piece of a wire of length l.

    1-give an expression for the total area of the square and circle formed.


    2-find the radius of the circle and side of the square that make their areas equal.


    3-find values of the radius of the circle and side of the square which give the largest and smallest total area.

    4-If insted of the circle another square is formed.Find the values of the sides of the square that yeild the largest and smallest total area.





    3. The attempt at a solution

    I solve (1) & (2) but I have difficulties on (3) &(4)


    1-give an expression for the total area of the square and circle formed.

    we assume that :

    length of wire=circumference of circle+perimeter of square


    http://www.0zz0.com" [Broken]


    Area of square= a^2

    *To know the length of side of square:

    perimeter of square=4*a


    a= perimeter/4

    http://www.0zz0.com" [Broken]

    Area of square= a^2

    =http://www.0zz0.com/realpic.php?s=8&pic=2009/05/08/18/425527447.jpg"


    Area of circle=pi*r^2

    To know the radius:

    r= circumference of circle / 2 PI

    Area of circle = http://www.up-00.com/" [Broken]


    Total Area=Area of square +Area of circle


    =http://www.0zz0.com" [Broken]



    ===========================================

    2-find the radius of the circle and side of the square that make their areas equal.


    Area of square=Area of circle :





    http://www.0zz0.com/realpic.php?s=8&pic=2009/05/08/19/917299630.jpg"


    The value of radius and side of square :


    http://www.0zz0.com" [Broken]



    http://www.gulfup.com/ [Broken]
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 8, 2009 #2
    (3) Just add the areas, then differentiate it with respect to 'x' and set it equal to 0, and solve for 'x'. That will get either the max or min. Then check the limits (x=0 or x=L). Then you will have both the max and min.

    (4) Do the same as (3) but now use the formula for total area of 2 squares.
     
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