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6 July, 23:05

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 500 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.07 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

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  1. 7 July, 00:21
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    minimum cost is 15.51 cent

    Explanation:

    given data

    volume = 500 cubic centimeter

    side cost = 0.04 cents per square centimeter

    top costing = 0.07 cents per square centimeter

    to find out

    minimize production cost

    solution

    first we find the volume that is

    volume = π*r²*h

    so here put value and find h in r form

    500 = π*r²*h

    h = 500 / π*r² ... a

    so cost of packing will be

    cost = area of side + volume side + volume top

    cost = 2π*r*h*0.04 + π*r²*0.04 + π*r²*0.07

    here h from equation a

    cost = 2π*r * (500 / π*r²) * 0.04 + π*r²*0.04 + π*r²*0.07

    cost = 40 / r + 0.346*r² ... b

    for minimum put it equal zero and differentiate it

    -40 / r² + 0.691 * r = 0

    r = 3.86

    so from equation a

    h = 500 / π*r²

    h = 500 / π*3.86²

    h = 10.63 cm

    so total cost will be by equation b

    cost = 40 / 3.86 + 0.346*3.86²

    cost = 15.51 cent
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