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3 January, 16:26

A soup company wants to manufacture a can in the shape of a right circular cylinder that will hold 500 cm^3 of liquid. The material for the top and bottom costs 0.02 cent/cm^2, and the material for the sides costs 0.01 cent/cm^2.

a) Estimate the radius r and height h of the can that costs the least to manufacture. [suggestion: Express the cost C in terms of r.]

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  1. 3 January, 17:57
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    The area of the top and bottom:

    2πr²

    Cost for top and bottom:

    2πr² x 0.02

    = 0.04πr²

    Area for side:

    2πrh

    Cost for side:

    2πrh x 0.01

    = 0.02πrh

    Total cost:

    C = 0.04πr² + 0.02πrh

    We know that the volume of the can is:

    V = πr²h

    h = 500/πr²

    Substituting this into the cost equation to get a cost function of radius:

    C (r) = 0.04πr² + 0.02πr (500/πr²)

    C (r) = 0.04πr² + 10/r

    Now, we differentiate with respect to r and equate to 0 to obtain the minimum value:

    0 = 0.08πr - 10/r²

    10/r² = 0.08πr

    r³ = 125/π

    r = 3.41 cm
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