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Math Help - Optimization

  1. #1
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    Optimization

    I should know this, but my mind completely blanked out on me, and I am lost...

    A buyer wants lots of boxed of the same size with strong, square bases, and no tops. A manufacturer can make 1000 of them for a total of $36.00, charging 0.3 cents per square inch for the base, and 0.1 cents per square inch for the sides. The buyer wants to know how to choose the dimensions of the box so that it holds as much as possible, i.e., he wants to maximize the volume. Let x denote the edgelength of the base of the box and let y denote the height.

    Express the volume V of the box as a function of x alone.

    What is the domain of V given the fact that V represents a volume?

    Find the value of x in domain V which gives the maximum volume of V and what is the corresponding value of y?

    What is the range of V(x) for x in domain V? Explain how this range is related to the maximum value of V.


    Ok, so i began to set up an equation and got:
    4(.1xy)+.3(x^2)=.036
    therefore y=(3.6-.3(x^2))/(.4x)
    and after plugging that into the equation V=(x^2)(y)
    I got V=(.9x-.75(x^3)) but this is not correct.

    Can I get some input on what I need to do correctly, and what I have done wrong?

    Thank you in advance.
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  2. #2
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    Quote Originally Posted by mjoconn View Post
    I should know this, but my mind completely blanked out on me, and I am lost...

    A buyer wants lots of boxed of the same size with strong, square bases, and no tops. A manufacturer can make 1000 of them for a total of $36.00, charging 0.3 cents per square inch for the base, and 0.1 cents per square inch for the sides. The buyer wants to know how to choose the dimensions of the box so that it holds as much as possible, i.e., he wants to maximize the volume. Let x denote the edgelength of the base of the box and let y denote the height.

    Express the volume V of the box as a function of x alone.

    What is the domain of V given the fact that V represents a volume?

    Find the value of x in domain V which gives the maximum volume of V and what is the corresponding value of y?

    What is the range of V(x) for x in domain V? Explain how this range is related to the maximum value of V.


    Ok, so i began to set up an equation and got:
    4(.1xy)+.3(x^2)=.036
    therefore y=(3.6-.3(x^2))/(.4x)
    and after plugging that into the equation V=(x^2)(y)
    I got V=(.9x-.75(x^3)) but this is not correct.

    Can I get some input on what I need to do correctly, and what I have done wrong?

    Thank you in advance.
    one box costs 3.6 cents ... 0.036 would be in dollars

    0.3x^2 + 0.1(4xy) = 3.6

    3x^2 + 4xy = 36

    y = \frac{36-3x^2}{4x}

    V = x^2y

    V = \frac{36x-3x^3}{4}
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  3. #3
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    Quote Originally Posted by skeeter View Post
    one box costs 3.6 cents ... 0.036 would be in dollars

    0.3x^2 + 0.1(4xy) = 3.6

    3x^2 + 4xy = 36

    y = \frac{36-3x^2}{4x}

    V = x^2y

    V = \frac{36x-3x^3}{4}

    Thank you so much. Would the domain then be (0,infinity)?
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  4. #4
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    Quote Originally Posted by mjoconn View Post
    Thank you so much. Would the domain then be (0,infinity)?
    domain of V(x) ...

    \frac{36x-3x^3}{4} > 0

    3x(12-x^2) > 0

    3x(\sqrt{12}-x)(\sqrt{12}+x) > 0

    0 < x < \sqrt{12}
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