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Thread: Cola Can Project

  1. #1
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    Cola Can Project

    In my Ap Caluculus class we are on the chapter of derivative applications and we have this project to di.

    A right circular cylinder is to be designed to hold 12 fluid ounces of a soft drink and to use a minimum amount of material in construction. Find the required dimensions for the container. (1 fl. oz. = 1.80469 inches cubed)

    If someone could please help to get me started on this problem I would be very grateful.
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  2. #2
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    Quote Originally Posted by Vigo
    In my Ap Caluculus class we are on the chapter of derivative applications and we have this project to di.

    A right circular cylinder is to be designed to hold 12 fluid ounces of a soft drink and to use a minimum amount of material in construction. Find the required dimensions for the container. (1 fl. oz. = 1.80469 inches cubed)

    If someone could please help to get me started on this problem I would be very grateful.
    The amount of material required is proportional to the
    surface area of the cylinder. This is:

    $\displaystyle Area\ =\ (2.\pi.r).h\ +\ 2.(\pi.r^2)$

    that is the area of the curved part of the can, plus the area of the
    two end pieces.

    The volume of the cylinder is:

    $\displaystyle Volume\ =\ \pi.r^2.h$.

    Where $\displaystyle r$ is the radius of the base, and $\displaystyle h$ is the height of
    the can.

    That the volume is fixed allows you to write $\displaystyle h$ as a function
    of $\displaystyle r$ by rearranging the second of these equations. Substituting
    this for $\displaystyle h$ in the first equation gives you the surface area
    as a function of $\displaystyle r$ and $\displaystyle Volume$.

    You then need to find the minimum of $\displaystyle Area$ by
    differentiating w.r.t. $\displaystyle r$, and setting the derivative
    equal to $\displaystyle 0$.

    RonL
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  3. #3
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    OK so:

    h = 12/(pi*r^2)

    and

    f(x) = (2*pi*r)*(12/pi*r^2) + 2(pi*r^2)

    Find the derivative of f(x) and set that equal to 0.

    Is all of this right?
    If it is, what does the final answer tell you?
    And where does the 1 fl. oz. = 1.80469 inches cubed come into this problem?
    Thanks again.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Vigo
    OK so:

    h = 12/(pi*r^2)

    and

    f(x) = (2*pi*r)*(12/pi*r^2) + 2(pi*r^2)

    Find the derivative of f(x) and set that equal to 0.

    Is all of this right?
    If it is, what does the final answer tell you?
    And where does the 1 fl. oz. = 1.80469 inches cubed come into this problem?
    Thanks again.
    If you use inches as you unit of linear measure (i.e. for h and r), then
    you should use cubic inches as your unit of volume. So you should
    have:

    h = (12*1.80469)/(pi*r^2),


    RonL
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  5. #5
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    Alright thanks a lot for all of the help. I am going to work on this and post my final answers soon.
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