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Homework Help: Surface Area of A Can

  1. Aug 30, 2011 #1
    1. The problem statement, all variables and given/known data
    Is minimiznig the area of tin used to make a can an important factor?
    Suppose a manufacturer wishes to enclose a fixed volume,V, using a cylindrical can.
    The height of the cylinder is denoted by h, and the radius of the cylinder can section by r.
    i)Write a function for the surface area of the can.
    ii)Determine what happens to the surface area as the radius increases.
    iii)Determine what happens as the radius tends to zero.
    iv)Find the values of r which minimizes the surface area.
    v)Consider an alternative tin shape, with justifications, does your result support the argument that minimizing surface area is a key factor in the design of tin cans?

    2. Relevant equations
    Volume = πr^2h

    3. The attempt at a solution

    i)Surface Area = 2πr2 + 2πrh = 2πr(r+h)
    ii)The surface area increases and the height decreases as the radius increases.

    Confused about the rest.
    Last edited: Aug 30, 2011
  2. jcsd
  3. Aug 30, 2011 #2
    The answer is 42.
  4. Aug 30, 2011 #3
    care to explain how you arrived at 42? thanks
  5. Aug 30, 2011 #4


    Staff: Mentor

    You also know that the volume V is fixed, so you can solve for r as a function of h and the fixed V.
  6. Aug 31, 2011 #5
    do you mean like 2πr^2 + 2πr(V ÷ πr^2) and differentiate?
  7. Aug 31, 2011 #6


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    Yes, now you have the surface area in terms of the single variable r. Now you can minimize it using derivatives.
  8. Aug 31, 2011 #7
    when i did the differentiation i got r^3 = 2V/4π

    Is this correct?
  9. Aug 31, 2011 #8


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    That agrees with what I got as the answer to iv). Though you haven't actually solved for r yet.
  10. Aug 31, 2011 #9
    iv) i thought i was answering iii) :confused:
    so how do i get the answer for iii) ?
  11. Aug 31, 2011 #10


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    You gave an answer to a question. If the question is "iii)Determine what happens as the radius tends to zero." I don't think "r^3 = 2V/4π" makes much sense as an answer. If the question is "iv)Find the values of r which minimizes the surface area." Then I think it does. Are you trying to confuse me?
  12. Aug 31, 2011 #11
    yea sorry about that i now understood that i actually answered iv)
    not trying to confuse you , just a lapse in concentration on my part.
    thanks for your help btw
  13. Sep 1, 2011 #12
    When I responded to the OP, it was just the blank homework form. You must have come back and filled it in later.
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