Question

Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0. 1 cm thick and the metal in the sides is 0. 05 cm thick.

Answer 1

The amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal on the top and the bottom is 0.1 cm thick and the metal on the sides is 0.05 cm thick is 8.8 cm.

The formula for calculating the volume of a cylinder is given below.

V = πr^2 h

Get the differential of the volume as shown:

dV = V/ h dh + V / r dr

V/ h = πr^2

V/ h  = 2  πr h

Now, the differential becomes

dV =  πr^2dh +  2πrh dr

Given the following parameters i.e. diameter and height

dh = 0.1 + 0.1 = 0.2 cm

dr = 0.05 cm

h = 10 cm

d = 4 cm  

r = 2cm

Substituting the values in the above equation, we get

dV = 3.14(2)^2(0.2)  + 2(3.14)(2)(10)(0.05)

dV = 2.512 + 6.28

dV = 8.792 cm

dV = 8.8 cm

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