How would these questions concerning consumer theory be accessed?

3) Suppose the utility function U(x,y) = x"y" represents my preferences. Is it possible to write down another utility function that represents the same preferences? If so, write down two other utility functions and explain what must be true in order for those functions to represent the same preferences. 4) Joyce gets enjoyment from two things - skydiving and white water rafting. The following table lists various consumption bundles that are composed of these activities, with a blank column for the total utility she obtains from each bundle: Bundle # Skydiving Trips # Rafting Trips Total Utility A 8 B 8 1 C 2 D 16 E 4 4 a. Suppose that one possible representation of Joyce's preferences is given by the utility function: U(S, R) = 5S2/3R1/3. Given this utility function, fill in the column for Joyce's total utility associated with each bundle. b. Using the preference ranking notation, rank the following bundles. Also state what the notation means in each case. i. A and B ii. A and C c. Suppose that Joyce has standard preferences that satisfy all usual assumptions. Without the utility values listed on the table, would you be able to rank any of the following pairs of bundles? If so, how would you rank them and why? i. A and B ii. A and C iii. C and E d. On a graph, use the information in the table to plot all of the commodity bundles. Draw a set of indifference curves that would be consistent with the information given. Label all axes, commodity bundles, and indifference curves on the graph