Use differentials to estimate the amount of material in a closed cylindrical can

Question

Use differentials to estimate the amount of material in a closed cylindrical can that is 70 cm high and 28 cm in diameter if the metal in the top and bottom is 0.2 cm thick, and the metal in the sides is 0.05 cm thick. Note, yon are approximating the volume of metal which makes up the can (i.e. melt the can into a blob and measure its volume), not the volume it encloses. The differential for the volume is dV = __________ dr + __________ dh dr = __________ and dh = __________ (be careful) The approximate volume of material is __________ cm^3

Explanation / Answer

V = r2h , r=28/2 = 14 cm , h=70cm

dV = (V/r)dr + (V/h)dh

dV = (2rh)dr + (r2)dh

Now , dr=0.05 cm , dh = 2*0.2 = 0.4 cm

then , dV = 2(14)(70)(0.05) + (14)2(0.4) = 98 + 78.5 = 176.4

dV = 554.177 cm3

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