This question about maximum error is giving me a hard time, you can see what I tried to do to solve it just below.
Q: The height and radius of a circular cylinder are measured with possible errors of 2% and 4%, respectively. Aproximate the maximum error percetange in calculating the cylinder volume.
I've tried using dV here, pluggin in the values 0,02 and 0,04 and then $$ P = \frac{dV}{V} $$ to get to the result but it's wrong. I don't know what else to try as this particular section is new to me. Help would be greatly appreciated.
Edit: the correct answer is 10%
Best Answer
The volume of a cylinder is $V=\pi r^2h$. Then the relative error is
$$\frac{dV }V=\frac{\frac{dV}{dr}\cdot \Delta r+\frac{dV}{dh}\cdot \Delta h}{V}$$
The numerator on the right hand side is the total differential.
$$\frac{dV }V=\frac{2\pi rh\cdot \Delta r+\pi r^2\cdot \Delta h}{\pi r^2h}$$
$$\frac{dV }V=\frac{2\pi rh\cdot \Delta r}{\pi r^2h}+\frac{\pi r^2\cdot \Delta h}{\pi r^2h}$$
Cancelling out
$$\frac{dV }V=\frac{2\cdot \Delta r}{r}+\frac{\Delta h}{h}$$
The relative errors of the radius and the height are $\frac{ \Delta r}{r}$ and $\frac{\Delta h}{h}$, respectively.