$3$ points$)$ What is the short run cost function? Show the quantity the firm would want to produce the same as in part $($a$).$

$($c$)(3$ points$)$ Suppose a manager finds a new management technique that would produce output using the following production function:

q~$($K$,$L$)=20($K$+$L$)$

Is the firm better off or worse off switching to the new management technique? $(\backslash$textit$\{$hint$\}$: Compare short run profits.$)$

$($d$)(6$ points$)$ In the long run both capital and labour can be adjusted. Assuming the firm is producing q$=300$ using the new technique $($i$.$e$.$ q~$),$ what is the optimal amount of capital and labour hired? Does the firm make a profit here? How does profit compare with part $($c$)?$

$($e$)(2$ points$)$ Find the long run cost curve for q~$.$