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

Question

Use differentials to estimate the amount of metal in a closed cylindrical can that is 22 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. () The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV = 8.31 T, where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the volume increases from 10 L to 10.6 L and the temperature decreases from 345 K to 335 K. (Note whether the change is positive or negative in your answer.)

Explanation / Answer

Solution:(24)

V is a function of two variables, r and h, we have: V= r2h

The total differential of V is dV:

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

V/r = 2rh (differentiate V with respect to r, keeping h constant)
V/h = r2 (differentiate V with respect to h, keeping r constant)

So, we now have:

dV = (V/r).dr + (V/h).dh = 2rh.dr + r2.dh

h = 22cm
r = d/2 = 10/2 = 5cm
dr = 0.05cm
dh = 2 x 0.1cm (Thickness of metal in top and bottom)

dV = 2(5)(22)(0.05) + (5)2(2x0.1)

dV = 16 cm3 50.265 cm3

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