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Math Help - optimzation problem #9

  1. #1
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    optimzation problem #9

    A cylindrical-shaped tin can is to have a capacity of 1000 cm^3.
    a) find the dimensions of the can that require the minimum amount of tin and assume that no waste material is allowed. The marketing department has specified that the smallest can the market will accept has a diameter of 6 cm and a height of 4 cm.
    b) express the answer for part a as a ratio of height to diameter

    so far.... I have
    (not sure if it is correct)

    V=1000cm^3

    (0.5d)^2+(0.5h^2)=r^2

    V=pir^2h
    1000=[pi][(0.5d)^2+(0.5h)^2][h]
    Last edited by mr fantastic; August 12th 2009 at 03:49 PM. Reason: Changed post title
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    A cylindrical-shaped tin can is to have a capacity of 1000 cm^3.
    a) find the dimensions of the can that require the minimum amount of tin and assume that no waste material is allowed. The marketing department has specified that the smallest can the market will accept has a diameter of 6 cm and a height of 4 cm.
    b) express the answer for part a as a ratio of height to diameter

    so far.... I have
    (not sure if it is correct)

    V=1000cm^3

    (0.5d)^2+(0.5h^2)=r^2 this is not a correct relationship ... r is not the distance from one corner of the can to the other.

    V=pir^2h
    1000=[pi][(0.5d)^2+(0.5h)^2][h]
    1000 = \pi r^2 h

    h = \frac{1000}{\pi r^2}

    what will be minimized is the surface area ...

    S = 2\pi r^2 + 2\pi rh

    S = 2\pi r^2 + 2\pi r \cdot \frac{1000}{\pi r^2}

    S = 2 \pi r^2 + \frac{2000}{r}

    find  \frac{dS}{dr} and minimize.
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  3. #3
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    Quote Originally Posted by skeeter View Post
    1000 = \pi r^2 h

    h = \frac{1000}{\pi r^2}

    what will be minimized is the surface area ...

    S = 2\pi r^2 + 2\pi rh

    S = 2\pi r^2 + 2\pi r \cdot \frac{1000}{\pi r^2}

    S = 2 \pi r^2 + \frac{2000}{r}

    find  \frac{dS}{dr} and minimize.
    ok.. question, would the domain of r be..
    3<=r<=1000?

    Im not sure about how to find the largest possible radius
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  4. #4
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    Quote Originally Posted by skeske1234 View Post
    A cylindrical-shaped tin can is to have a capacity of 1000 cm^3.
    a) find the dimensions of the can that require the minimum amount of tin and assume that no waste material is allowed. The marketing department has specified that the smallest can the market will accept has a diameter of 6 cm and a height of 4 cm.
    b) express the answer for part a as a ratio of height to diameter
    based on what the marketing department says ...

    r \ge 3 cm and h \ge 4 cm

    since h \ge 4 ...

    \frac{1000}{\pi r^2} \ge 4

    \frac{1000}{4\pi} \ge r^2

    r \le \sqrt{\frac{250}{\pi}}

    so ...

    3 \le r \le \sqrt{\frac{250}{\pi}}
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