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Homework Help: Optimization: Minimum Surface Area

  1. Jul 19, 2008 #1
    1. The problem statement, all variables and given/known data

    I need help on an optimization problem involving a hexagonal prism with no bottom or top, but the top is covered by a trihedral pyramid which has a displacement, x, such that the surface area of the object is at a minimum for a given volume. The assigned variables include:
    A= the surface area of the object
    a= the length of each side of the hexagon (a=1)
    h= the height of the prism
    x= displacement

    2. Relevant equations

    Given that A= 6(ah-1/2ax) + 3a[tex]\sqrt{3}[/tex]*[tex]\sqrt{x^2+(a^2/4)}[/tex], use calculus to find the displacement, x that yields the minimum surface area. Calculate x if a=1.


    3. The attempt at a solution

    I'm struggling trying to find a secondary equation for this optimization problem and I think I need to use an equation evolving h, but I'm not sure what.
     
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  3. Jul 19, 2008 #2

    arildno

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    The volume V is to be constant, this will lead you to an equation in h and x, from which you may find an expression for h.
     
  4. Jul 20, 2008 #3
    How would I use a formula for volume since the problem works with two shapes.

    V=(3[tex]\sqrt{3}[/tex])/2 a^2*h

    h= V/ (3[tex]\sqrt{3}[/tex]/2 a^2*h

    Then, I can just plug this into the first equation, simplify, and start the calculus. Is that right?
     
  5. Jul 20, 2008 #4

    arildno

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    Your TOTAL volume is the sum of the prism's volume and the pyramid's volume.

    That is the equation in h and x you are to set up!
     
  6. Jul 20, 2008 #5

    arildno

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    Your total volum is therefore:
    [tex]V=\frac{3\sqrt{3}}{2}a^{2}h+\frac{1}{3}\frac{3\sqrt{3}}{2}a^{2}x\to{h}=\frac{2V}{3\sqrt{3}a^{2}}-\frac{x}{3}[/tex]
     
  7. Jul 20, 2008 #6
    So after substituting that in for h, the simplified form would be:

    A= 4V-5x[tex]\sqrt{3}[/tex]+ 9[tex]\sqrt{x^2+4}[/tex]

    [tex]\sqrt{3}[/tex]
     
  8. Jul 20, 2008 #7
    How would I take the derivative of this equation then with multiple variables?
     
  9. Jul 21, 2008 #8

    arildno

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    You only have one variable, x. Remember, V is a constant!
     
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