Consider the following short$-$run production function $($where L $=$ variable input labor hours; Q $=$ output units produced$)$:$$

Q$=10$L $0\mathrm{.}5$L$2$

Suppose that each unit of output can be sold for $$10$ per unit. $$Also assume that the firm can obtain as much of the variable input $($L$)$ labor as it needs at $$20$ per hour.$$

Determine the marginal revenue product function.$$

Determine the marginal factor cost function.$$

Determine the optimal value of L$,$ given that the objective is to maximize profits.$$