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Cola Can Project

  1. Dec 6, 2005 #1
    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.
     
  2. jcsd
  3. Dec 6, 2005 #2
    [tex]\text{SA}=2\pi r\left(h+r\right)[/tex]

    [tex]\text{V}=\pi r^{2}h=12[/tex]

    I suggest you find an expression for h in terms of r in the second and plug it into the first to simplify. The rest should be easy assuming you know how to optimize a function using differential calculus.
     
  4. Dec 6, 2005 #3

    Galileo

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    Translate the problem into a mathematical problem.
    You want to minimize the surface area of a cylinder under the constraint that the volume must be some given volume V.

    So start by drawing a picture. Introduce variables that will be important (height, radius, suface area). And relations between them. This is the given. What is the unknown?
     
  5. Dec 7, 2005 #4
    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.
     
  6. Dec 7, 2005 #5

    Galileo

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    Okay, so f is the surface area of the can as a function of the radius r (not x!). You can minimize this using whatever knowlegde you have of calculus (=yes, differentiating would be the standard procedure).

    The "1 fl. oz. = 1.80469 inches cubed" comes from the conversion from some archaic unit such as fluid ounce to another one, namely the inch^3. You should use whatever dimensions you are comfy with and do the proper conversion.
     
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