Dudley's utility function is U$($C$,$R$)=$C$(12$R$)^2,$ where R is the amount of leisure he has per day. He has $16$ hours a day to divide between work and leisure. He has an income of $$20$ a day from nonlabor sources. The price of consumption goods is $$1$ per unit. $($a$)$ If Dudley can work as many hours a day as he likes but gets zero wages for his labor, how many hours of leisure will he choose?b$)$ If Dudley can work as many hours a day as he wishes for a wage rate of

$$10$ an hour, how many hours of leisure will he choose? How many hours will he work? $($Hint: Write down Dudley's utility subject to this constraint. Rement of leisure that maximizes his wishes to supply ine amount of labor he wis demand for leisure.$)($c$)$ If Dudley's nonlabor income decreased to $$5$ a day, while his wage rate remained at $$10,$ how many hours would he choose to work?Suppose that Dudley has to pay an income tax of $20$ percent on all of his income, and suppose that his before$-$tax wage remained at $$10$ an hour and his before$-$tax nonlabor income was $$20$ per day. How many hours would he choose to work?