Question $4.(15$ pts$.)$ A U$.$S$.$ firm is considering to optimize its production. It's production function is $\backslash ($ q$=5$ L$^\{0\mathrm{.}75\}$ K$^\{0\mathrm{.}25\}\backslash ),$ where q is the output, L is the number of labor, and K is the number of capital employed. The prices of labor and capital are $\backslash ($ w$=5\backslash )$ and $\backslash ($ r$=10\backslash ).$

a$)$ Find the equation of the firm's isoquant with output level of $100.$ Draw it on a clearly labelled graph. $(3$ pts$.)$

b$)$ What is the long run expansion path? Find the equation $($condition$)$ that explains the expansion path and draw it$.(5$ pts$.)$

c$)$ If the capital is fixed at $\backslash (\backslash$mathrm$\{$K$\}=10\backslash ),$ what is the short run expansion path? Find the equation $($condition$)$ that explains the expansion path and draw it$.(3$ pts $.)$

d$)$ How much does it cost to product $100$ units of output in the long run? How about with capital fixed at $\backslash (\backslash$mathrm$\{$K$\}=10\backslash )?$ Make sure to draw the relevant isoquant$($s$),$ and isocost$($s$).(4$ pts$.)$