A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 cubic centimeters of soup. The

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 cubic

centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.03 cents per square

centimeter. The top will be made of glued paper, costing 0.08 cents per square centimeter. Find the dimensions for

the package that will minimize production cost.

Helpful information:

To minimize the cost of the package:

Radius:

Height:

Minimum

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Piper 4 days 2021-10-10T22:15:21+00:00 1 Answer 0

Answers ( )

    0
    2021-10-10T22:16:26+00:00

    Answer:

    x = r = 3,12 cm

    h = 11,45 cm

    C(min) = 10,09 $

    Step-by-step explanation:

    Let x be radius of base  and  h height of cylinder

    Let A₁ be   Cylinder area of base  +  lateral area

    A₁  =  π*x²  +  2*π*x*h        Now V = 350 cm³     350  = π*x²*h

    h = 350/ π*x²

    Then

    A₁  =  π*x²  +  700/x

    A₂  area of the top

    A₂  = π*x²

    Now we write the expression of cost as fuction of x

    C(x)  = 0,03 * ( π*x²  +  700/x)   +  0,08*π*x²

    C(x)  = 0,03* π*x²  +   21/x   +  0,08*π*x²

    C(x)  = 0,11* π*x² +   21/x  

    Taking derivatives on both sides of the equation

    C´(x)  =  2*0,11*π*x  – 21 / x²

    C´(x)  = 0         0,22*π*x  – 21 / x²   =  0

    0,22*π*x³  –  21  =  0         0.6908* x³  –  21   = 0

    X³ =  21/0,6908               X³ =  30,40

    x  =  3,12 cm

    And

    h  =  350/π*x²        h  =  350 / 30,57        

    h = 11,45 cm

    C(min)  =  0,11* π*(3,12)² +   21/3,12  

    C(min)  =  3,36  + 6,73

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