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Math Help - Problem with optimising

  1. #1
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    Problem with optimising

    Hi there, I got a school problem that I need solving, I don't want to get the answer but I would love to get pointer as to where I have done something wrong.

    The problem is as following:

    I got a box with a lid on top. The base area is X*X and the height is H. H is the box + the extra height of the lid.

    The lid has the size of the base area + 4 side plates that are 3 cm high.

    The conditions are that the box should be 1000kubic cm large and the material needed should be as little as possible.


    I figured that if I am to decide what the least amount of materials needed I need to set up a function and I have decided to calculate the Area of materials as a function of X.

    With the conditions that X^2*(H-3)=1000 I get that H=1000/X^2+3

    Total amount of materials f(X)=2X^2+4HX F(X)=2X^2+4*(1000*X^-2+3)X
    F(X)=2X^2+(4000X^-2+13)X
    F(X)=2X^2+4000X^-1+13X

    And by derivation I should find the dimensions that gives us the least materials used. It feels wrong but I don't know what it is. So if you see any direct errors in the math please tell me and any pointers or hints as to how I should proceed would be greatly appreciated.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Rule of thumb

    Quote Originally Posted by dipsy34 View Post
    Hi there, I got a school problem that I need solving, I don't want to get the answer but I would love to get pointer as to where I have done something wrong.

    The problem is as following:

    I got a box with a lid on top. The base area is X*X and the height is H. H is the box + the extra height of the lid.

    The lid has the size of the base area + 4 side plates that are 3 cm high.

    The conditions are that the box should be 1000kubic cm large and the material needed should be as little as possible.


    I figured that if I am to decide what the least amount of materials needed I need to set up a function and I have decided to calculate the Area of materials as a function of X.

    With the conditions that X^2*(H-3)=1000 I get that H=1000/X^2+3

    Total amount of materials f(X)=2X^2+4HX F(X)=2X^2+4*(1000*X^-2+3)X
    F(X)=2X^2+(4000X^-2+13)X
    F(X)=2X^2+4000X^-1+13X

    And by derivation I should find the dimensions that gives us the least materials used. It feels wrong but I don't know what it is. So if you see any direct errors in the math please tell me and any pointers or hints as to how I should proceed would be greatly appreciated.
    This applies to all optimization problem I have ever found....you will have one condition lets say C_1=f(x,y)....then you have the optimization equation lets say O_1=f(x,y)....since C_1 is a constant you solve it for either x or y...imput that substition into O_1=f(x,y) to get either O_1=f(x) or O_1=f(y)...then find {O_1}' then by using standard techniques find the max or min..if that is confusing just say so and I will hlep with this specific problem....Mathstud
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  3. #3
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    It's a little confusing and I have been at this for the last few days since I really want to crack this nut but I have to throw in the towel and ask for some extra help.


    The conditions are that the box needs a volume of 1000 and that the base area is X^2 and the height h. Previously I said that the height was h-3 but after looking in to the assignment more closely I have changed my mind.

    The lid of the box has the area X^2+12x and the area of all the sides including the bottom of the box is X^2+4hx.

    I just cant figure out what to do in the next step, I tried to set that h=1000/X^2 but when I did that my graphs went insane and not producing any values I could use so clearly this mustn't be the way to solve the problem or it could be as simple as me making an error somewhere.


    Could really use some of that extra help Mathstud28
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  4. #4
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    earboth's Avatar
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    Quote Originally Posted by dipsy34 View Post
    ...
    The problem is as following:

    I got a box with a lid on top. The base area is X*X and the height is H. H is the box + the extra height of the lid.
    The lid has the size of the base area + 4 side plates that are 3 cm high.
    The conditions are that the box should be 1000kubic cm large and the material needed should be as little as possible....

    With the conditions that X^2*(H-3)=1000 I get that H=1000/X^2+3

    ...
    Let a denote the total amount of material. Then you know:

    a = x^2+4xH+4\cdot 3x + x^2 Substitute H =\frac{1000}{x^2} + 3 and you have your function "amount of material" with respect to x:

    a(x) = 2x^2+12x+\frac{1000}{x^2} + 3

    Calculate the first derivative:

    a'(x) = 4x+12-\frac{8000}{x^3}

    Now solve for x: a'(x) = 0

    I don't know how to solve this equation algebraically. But my computer says that the positive real solution is x \approx 6.04664...
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