In the calculation of the volume of a cube of nominal size $5$ inches, the uncertainty in the measurement of each side is $10\%.$ Find the uncertainty in the measurement of the volume.
I tried solving the problem as follows:
Given, the nominal size of the cube is $5$ inches. So, if the side length of the cube is $a$ then, $a\approx 5$ inches. Now, the uncertainty in measurement of each side is $10%$ i.e $5\times 10\%=0.5$ inches. Thus, the measured side length of the cube , is $a\pm 0.5$ and the maximum length of the cube possible, is $5+0.5=5.5$inches. So, the maximum volume is, $5.5^3=166.375$ .
Now, the nominal volume is, $a^3=5^3=125$ and the maximum absolute error possible in the measurement of volume, is $166.375-125=41.375$ due to which, the relative error in the measurement of volume, is $41.375/125=0.331$. Now, the percentage error is indeed, $0.331\times 100=33.1$. The uncertainty in the measurement of volume is actually the percentage error, so the uncertainty in volume is $33.1\%$ as well.
However, the answer given is $30%.$ I am not sure whether I made a mistake.
The probability of me making a mistake is possible because, I just felt that calculating the uncertainty in volume, means calculating the percentage error in the measurement of volume. I made this conclusion just intuitively.
Next, while calculating the percentage error I needed to calculate the relative error. To calculate the relative error, I needed to calculate the absolute error and to find the absolute error I needed the measured volume. But then, I just felt that I should take the measured volume to be the maximum possible volume. That is the reason why, I chose the measured side length of tge cube to $5.5$ inches to obtain the maximum possible volume. But all these things, I did was just me following my intuitions.
I don't really understand whether all these steps are justified in here.