Sorry for asking 5 questions at once! The first four are free response and the last one is multiple choice with the options below the question. Thanks in advance :)

1) Imagine there are two goods: cabbages and rice. Your utility function is U = C^1/3R^1/4, here C is the number of cabbages you consume, and R is the bowl of rice you consume. The price of each kilo of cabbages is $10, and the price of each bowl of rice is $2. The budget is $50. What is the optimal quantity of cabbages to consume?

2) Imagine there are two goods: mooncakes and rice-dumpling. Your utility function is U = M²R, where M is the number of mooncakes you consume, and R is the number of rice-dumpling you consume. The price of mooncake is $5, and the price of rice-dumpling is $4. The budget is $48. What is the optimal quantity of the mooncake to consume?

3) Imagine there are two series of science-fictions:The Three-Body Problem(T) andGalactic Empires(G).Your utility function is U = TG, where T is the number of books you consume forThe Three-Body Problem, and G is the number of books you consume forGalactic Empires.If your budget is $200, the price of each book ofThe Three-Body Problem(T) is $50, and the price of each book ofGalactic Empires(G) is $100, how many books ofThe Three-Body Problem(T) will you consume?

4) Imagine there are two series of science-fictions:The Three-Body Problem(T) andGalactic Empires(G).Your utility function is U = TG, where T is the number of books you consume forThe Three-Body Problem, and G is the number of books you consume forGalactic Empires.If your budget is $200, the price of each book ofThe Three-Body Problem(T) is $50, and the price of each book ofGalactic Empires(G) is $100, how many books ofGalactic Empires(G)will you consume?

5) Paulina's preferences can be represented by the utility function U = 3X²Y^0.75. Her budget is $130. The price of a unity of good X is $8, and the price of a unity of good Y is $3. What is her optimal bundle? The option are in the form (X, Y)

a. 12, 11.3333

b. 12.7273, 12.7273

c. 11.8182, 11.8182

d. 11.8182, 12.7273