Answer:
[tex]\Delta V = 39.584\,cm^{3}[/tex]
Step-by-step explanation:
The volume of the diameter is:
[tex]V = \frac{\pi}{4}\cdot D^{2}\cdot l[/tex]
The differential of this expression is:
[tex]\Delta V = \frac{\partial V}{\partial D} \cdot \Delta D + \frac{\partial V}{\partial L} \cdot \Delta L[/tex]
The partial derivatives are presented hereafter:
[tex]\frac{\partial V}{\partial D} = \frac{\pi}{2}\cdot D\cdot l[/tex]
[tex]\frac{\partial V}{\partial D} = \frac{\pi}{2}\cdot (12\,cm)\cdot (30\,cm)[/tex]
[tex]\frac{\partial V}{\partial D} \approx 565.487\,cm^{2}[/tex]
[tex]\frac{\partial V}{\partial l} = \frac{\pi}{4}\cdot D^{2}[/tex]
[tex]\frac{\partial V}{\partial l} = \frac{\pi}{4}\cdot (12\,cm)^{2}[/tex]
[tex]\frac{\partial V}{\partial l} \approx 113.097\,cm^{2}[/tex]
The thickness in the sides is related to the diameter, whereas the thickness in the top and bottom is related to the height. The estimated amount of material is:
[tex]\Delta V = (565.487\,cm^{2})\cdot (0.05\,cm) + (113.097\,cm^{2})\cdot (0.1\,cm)[/tex]
[tex]\Delta V = 39.584\,cm^{3}[/tex]
