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22 August, 21:14

You are to construct a€cylindrical can out of tin that encloses a volume of exactly 100 cubic inches. to three significant figures, which dimensions (r and h) will give you the tin can that requires the least amount of metal?

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  1. 22 August, 21:26
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    The formula for volume of cylinder is:

    V = π r^2 h

    since volume is 100 cubic inches, so:

    100 = π r^2 h

    Rewriting in terms of h:

    h = 100 / π r^2

    The least amount of metal means least amount of surface area needed. The surface area of cylinder is:

    A = 2 π r h + 2 π r^2

    insert the value of h in terms of r:

    A = 2 π r (100 / π r^2) + 2 π r^2

    A = 200 r-1 + 2 π r^2

    Take the 1st derivate and set to 0, dA/dr = 0:

    dA/dr = - 200 r-2 + 4 π r

    0 = - 200 r-2 + 4 π r

    200 / r^2 = 4 π r

    r^3 = 50 / π

    r = 2.52 inches

    So height h is:

    h = 100 / π r^2

    h = 100 / π (2.52) ^2

    h = 5.01 inches

    Answers:

    r = 2.52 inches

    h = 5.01 inches
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