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.)

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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