Question: The Emperor want to make a platinum ball. However, he suspected that he has been cheated by the alchemist again by mixing some copper to the platinum ball. The platinum ball was sent to the laboratory for dimension and mass measurements to determine if the platinum ball is fake. You first carefully measured the mass ofthe platinum ball at 20 \"C which was found to be 272.520 grams. You have then measured the dimension of the platinum ball at 20 C for 10 times and found the values of diameter as follows: Diameter (d, cm): 3.013, 3.012, 2.997, 3.002, 3.002, 3.006, 2.994, 2.995, 2.998, 3.004 lfthe measured mass was found well within the range of nominal mass (m) made of pure platinum in association with the measurement uncertainty, then the ball shall be made of pure platinum. From the literature, you found that the density (p) of pure platinum is 21.450 grams/cm3 at 20 \"C (with a semi-range of 0.020 gra ms/cm3, due to trace amount of impurity during refinery) with an estimated reliability of 33 % for the data. Density (p) is derived by dividing the mass (m) of an object by its volume (V). On the other hand the volume (V) of a sphere can be calculated bythe following equation: 1:613 Volume (V) = 7 (Please show all you calculation steps in the A4 answer sheet.) (a) Derive the model for the mass (m) of platinum ball using the above information. (b) Calculate the mean diameter (d) and the corresponding nominal mass (m) ofthe platinum ball if it was made of pure platinum. (c) Compute the sensitivity coefficients of d(cd) and p(cp) by the use ofpartfal derivative approach. (d) Complete the following table (shown on next page) by calculating the standard uncertainties ofd and p, the sensitivity coefficients (Ci) (part (c)), degree of freedoms (Vi) and combined uncertainty (uc) of mass. Parameter Nominal Sensitivity Standard Ci *u Degree of value coefficient (ci) uncertainty (ui) freedom (vi) ---_ ml 9' \"c = Veff = (e) Determine the expanded uncertainty of mass (with 95% confidence level) and report the nominal mass and its expanded uncertainty in standard wordings (format). (f) Based on your calculations, explain ifthe platinum ball was intentionally mixed with copper. (You may use a graph to explain your conclusion.)