A pressurized air tank is nominally at ambient temperature $(25C).$ Using ideal gas law, answer the following questions,

$($a$)$ How accurately can the density be determined if the temperature is measured with an uncertainty of $2C$ and the tank pressure is measured with a relative uncertainty of $1\%?$

$($b$)$ Suppose that the density determination must be within $0\mathrm{.}5\%$ to be useful. If the temperature measurement can be made to within $1C,$ how accurate must the pressure measurement be$?$

$($c$)$ In class, when we first discussed estimating the standard deviation of a sample population, we discussed the importance of the sample containing the influence of all the variations of interest. Comment on the time scales of factors that have a significant influence on the data and that contribute to the random uncertainty.