$1.$ Suppose that a firm's production function is given by: $\backslash (\backslash$mathrm$\{$Q$\}=\backslash$mathrm$\{$L$\}^\{0\mathrm{.}5\}\backslash$mathrm$\{$~K$\}^\{0\mathrm{.}5\}\backslash ).$ Suppose that the wage firms must pay for their workers is $\backslash (\backslash$$ $90\backslash )$ per worker and the rental rate they must pay for their capital is $\backslash (\backslash$$ $10\backslash )$ per unit.

a$.$ Calculate the firm's SR production function if the firm uses $36$ units of capital.

b$.$ Calculate the firm's SR total cost function $($must be as a function of Q $).$

c$.$ Suppose the firm wants to produce $18$ units of output, but it is the SR$,$ so they cannot alter the amount of capital they use. How much labor will they need to produce these $18$ units?

d$.$ What is the short$-$run total cost of producing these $18$ units.

e$.$ Calculate the cost$-$minimizing combination of L and K if the firm wishes to produce $18$ units of output when K is not fixed.

f$.$ What is the total cost of producing $18$ units in the long$-$run $($when K is flexible$).$ Compare your answer to part d$.$

g$.$ Calculate the LR total cost function $($must be a function of Q$).$ Double check your answer against your answer to f for $18$ units.