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6 February, 13:06

For a closed cylinder with radius r ⁢ cm and height h ⁢ cm, find the dimensions giving the minimum surface area, given that the volume is 40 cm3.

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  1. 6 February, 13:17
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    we know the surface area of the two ends of the closed cylinder is: 2 (r 2).

    The surface are of the side of the cylinder is h (2r)

    So the total surface area is: 2 (r 2) + h (2 r)

    The volume of the cylinder is r 2 h = 40 cm3

    now solve this last equation for h:

    so, h = 40 / (*r 2)

    Substitute the value for h into the expression for the total surface area:

    2 (r 2) + (40 / (r 2)) (2 r)

    Now Simplify the above expression we get:

    2 (r 2) + (80 / r)

    Now take the derivative:

    4 r - 80 r - 2

    Set it equal to zero and solve for r:

    4 r - 80 r - 2 = 0

    Divide by 4:

    *r - 20 r - 2 = 0

    Multiply by r2 we get:

    r3 - 20 = 0

    r3 = 20

    r3 = 20/

    Hence,

    r = (20 / ) (1/3)

    r 1.8534 cm

    Substitute this value for r into h = 40 / (r 2) and evaluate h:

    h = 40 / * (1.8534) 2

    h 3.7067
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