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Math Help - Calculus 3 Help needed-Minimizing

  1. #1
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    Calculus 3 Help needed-Minimizing

    I'm self teaching myself multivariable calculus and have gotten stuck on problems involving minimizing and maximizing. The two problems that I am currently stuck on are as follows.

    1)You are creating a retangular box with a volume of 32000cm^3. Find dimensions that minimize the material used in the construction of this box.

    2) Find three numbers whose sum add to 100 and whose product is a minimum.

    When I work both problems my answers come out as maximums and not minimums, assistance will be appreciated very much.

    Thank you
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  2. #2
    MHF Contributor arbolis's Avatar
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    Hi,
    When I work both problems my answers come out as maximums and not minimums
    Can you show us your answers?
    Did you use the Lagrange multipliers method?
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  3. #3
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    Quote Originally Posted by Reskal View Post
    I'm self teaching myself multivariable calculus and have gotten stuck on problems involving minimizing and maximizing. The two problems that I am currently stuck on are as follows.

    1)You are creating a retangular box with a volume of 32000cm^3. Find dimensions that minimize the material used in the construction of this box.

    2) Find three numbers whose sum add to 100 and whose product is a minimum.

    When I work both problems my answers come out as maximums and not minimums, assistance will be appreciated very much.

    Thank you
    I got the first to work out - show us what you tried. For the second, you'll need to try the boundary values (is zero allowed or do the numbers have to be positive?)
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  4. #4
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    The numbers must be positive for both problems and cannot be zero.

    For the first problem I set up the two equations

    xyz=32000
    2xz+2yz+xy=f(x,y,z)

    Solved for z=3200/xy
    and then set that in the second equation to get

    64000/y+64000/x+xy=f(x,y,z)

    taking the gradient I get (-64000/x^2+y, -64000/y^2+x)

    When I began to do the second derivative test and solved for fxx I realized that my answer would yield a maximum and was stuck there.

    (langrange multipliers is not until the next section so I doubt I am supposed to use those, will these start making these problems easier?)

    For the second problem

    I set the two equations to be x+y+z=100 and xyz=f(x,y,z)

    solved for z=100-x-y and subed that into the second equation to get

    xy(100-x-y)

    took the gradient of that to get

    fx=100y-2xy-y^2
    fy=100x-x^2-2xy

    and got stuck after that.

    Thanks all!
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  5. #5
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    Quote Originally Posted by Reskal View Post
    The numbers must be positive for both problems and cannot be zero.

    For the first problem I set up the two equations

    xyz=32000
    2xz+2yz+xy=f(x,y,z)

    Solved for z=3200/xy
    and then set that in the second equation to get

    64000/y+64000/x+xy=f(x,y,z)

    taking the gradient I get (-64000/x^2+y, -64000/y^2+x)

    When I began to do the second derivative test and solved for fxx I realized that my answer would yield a maximum and was stuck there.

    (langrange multipliers is not until the next section so I doubt I am supposed to use those, will these start making these problems easier?)

    For the second problem

    I set the two equations to be x+y+z=100 and xyz=f(x,y,z)

    solved for z=100-x-y and subed that into the second equation to get

    xy(100-x-y)

    took the gradient of that to get

    fx=100y-2xy-y^2
    fy=100x-x^2-2xy

    and got stuck after that.

    Thanks all!
    First is the box and open top (you didn't mention that). Second, factor f_x and f_y and look at case where each is zero.
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