$S=20L^{0\mathrm{.}30}{C}^{0\mathrm{.}70}$

In this formula $L$ represents the units of labor input and $C$ the units of capital input. Each unit of labor costs $$50,$ and each unit of capital costs $$100.$

$($a$)$ Formulate an optimization problem that will determine how much labor and capital are needed in order to produce $45,000$ tons of steel at minimum cost.

min

s$.$t$.$

$=45,000$

$L,C0$

and upper bound constraints of $0$ and $5,000$ for both $L$ and $C$ before solving. $($Round your answers to three decimal places.$)$

$

$at(L,C)=(,)$