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21 June, 07:14

Use differentials to estimate the amount of metal in a closed cylindrical can that is 30 cm high and 10 cm in diameter if the metal in the top and the bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick. (Round your answer to two decimal places.)

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  1. 21 June, 11:02
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    The amount of metal in the cylindrical can is 20π cm³

    Step-by-step explanation:

    By definition, the volume of a cylindrical can is

    V = πr²h

    Using the properties of Total Differential,

    dV = (đV/đr) dr + (đV/đh) dh

    Where đ represent partial differential.

    We are given the following parameters:

    Height, h = 30cm

    Diameter = 10cm

    Because Diameter is twice the radius,

    Radius = (10/2) cm = 5cm

    dr = 0.05

    dh = 2 (0.1) = 0.2 (This covers for the top and bottom)

    Differentiating V partially with respect to r, we have

    đV/đr = 2πrh

    Differentiating V partially with respect to h, we have

    đV/đh = πr²

    Now, we substitute these values into the equation:

    dV = (đV/đr) dr + (đV/đh) dh

    dV = [ (2π) (5) (30) ] (0.05) + (25π) (0.2)

    = 15π + 5π

    dV = 20π

    Which is what we want.
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