3. [20 pts] During a recent lab exam, students determined the density p of a metal cylinder. To find this, they used a laboratory scale to measure the mass m to within an experimental uncertainty of Am. They then used a micrometer to measure the diameter d to Ad, confirming that measurement with a dial caliper. Finally, they used the dial caliper to measure the length to Al. The resulting density is given by the equation P= 4m dl' Assume that the errors are independent; i.e., they are not correlated, and do not necessarily line up in either a worst-case or a best-case fashion. (a) [4 pts] Using the variables above, write down an algebraic equation for the ratio of the density uncertainty to the density, Ap/p. (b) [8 pts] Given the following measurements and uncertainty estimates: m = 59.812 g d = 0.9823 in 1 = 1.784 in Am 0.010 g Ad 0.0005 in Al = 0.005 in What is the absolute density uncertainty Ap (expressed in g/in) for these values? (c) [4 pts] Imagine that we could decrease the uncertainty any one of these measurements by a factor of ten. Which measurement do we most need to improve? Which measurement do we least need to improve? (d) [4 pts] What would be the new density uncertainty Ap (in g/in) after a tenfold decrease in the uncertainty of the measurement that most needs improvement?