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Homework Help: Determine the dimensions of a rectangular box, open at the top

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine the dimensions of a rectangular box, open at the top, having volume
    4 m3, and requiring the least amount of material for its construction. Use the
    second partials test. (Hint: Take advantage of the symmetry of the problem.)

    2. Relevant equations

    3. The attempt at a solution

    Volume, V= 4m^3
    let x = length
    y = width
    z = height

    4m^3 = xyz
    x = 4/yz

    because it is an open at the top rectangular box,
    the Surface, S = 2xz + 2yz + xy

    substitute x=4/yz inside the surface equation..
    S = 8/y + 2yz + 4/z

    to find the critical points, take the partial with respect to y and z.. the equal it to zero..

    S'(y)= 2z - 8/y^2
    S'(z)= 2y - 4/z^2

    solve the equations, i get,
    when y = 0, z = 0..
    y = 8, z = 1/16

    the problem is, what should i do next?
  2. jcsd
  3. Nov 23, 2009 #2


    Staff: Mentor

    When y = 0, 8/y^2 is undefined! When I set Sy and Sz to 0, I get different values than you did.

    Also, you're supposed to use the second partials test.
  4. Nov 23, 2009 #3
    ic... ok..


    S'(y) = 2z - 8/y^2
    S'(yy) = 16/y^3
    S'(yz) = 2

    S'(z) = 2y - 4/z^2
    S'(zz) = 8/z^3
    S'(zy) = 2

    what should i do next?
  5. Nov 23, 2009 #4


    Staff: Mentor

    You still haven't found the values of y and z for which Sy = 0 and Sz = 0.

    Also, you haven't used the second partials test; all you have done is calculate all the second partials.
  6. Nov 23, 2009 #5
    ok.. for.. S'(y)=0 -----> 2z - 8/y^2=0

    ......for.. S'(z)=0 -----> 2y - 4/z^2 =

    ok.. by letting z=4/y^2, i get y = 0, +8^1/2, -8^1/2

    am i got it correct now?
  7. Nov 23, 2009 #6
    is it possible if i use lagrange multiplier method?
  8. Nov 23, 2009 #7
    done with the equation for x, y and z..
    got some miscalculation just now..
    i got y=2, z= 1, x = 2..
    why i should do the second partial test?
  9. Nov 23, 2009 #8


    Staff: Mentor

    Those are the values I got.

    You should use the second partial test for two reasons:
    1. The problem asks you to use it.
    2. So you can tell whether these values give you a maximum value or a minimum value of S.
  10. Nov 23, 2009 #9
    thanks.. u are really helpful =)
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