WARNING: YOU MAY FIND THIS TO BE A PARTICULARLY CHALLENGING PROBLEM.
Suppose a Cobb-Douglas Production function is given by: $\displaystyle {P}{\left({L},{K}\right)}={22}{L}^{{0.2}}{K}^{{0.8}}.$ Furthemore, the cost function for a facility is given by the function:$\displaystyle {C}{\left({L},{K}\right)}={100}{L}+{400}{K}.$ Suppose the monthly production goal of this facility is to produce 15,000 items. In this problem, we will assume $\displaystyle {L}$ represents units of labor invested and $\displaystyle {K}$ represents units of capital invested, and that you can invest in tenths of units for each of these. What allocation of labor and capital will minimize total production Costs?

Hint 1: Your constraint equation involves the Cobb Douglas Production function, not the Cost function.

Hint 2: When finding a relationship between $\displaystyle {L}$ and $\displaystyle {K}$ in your system of equations, remember that you will want to eliminate $\displaystyle \lambda$ to get a relationship between $\displaystyle {L}$ and $\displaystyle {K}$.

Hint 3: Round your values for $\displaystyle {L}$ and $\displaystyle {K}$ to one decimal place (tenths).

Units of Labor, $\displaystyle {L}$ =

Units of Capital $\displaystyle {K}$ =

Also, what is the minimal cost to produce 15,000 units? (Use your rounded values for $\displaystyle {L}$ and $\displaystyle {K}$ from above to answer this question.)

Answer = $ (do not include any commas)