Question $1(20$ points$)$ In a perfectly competitive market, there are $20$ identical firms. Each of these firms has a total cost function of TC $=30+0\mathrm{.}2$q $2,$ where q is the firm$$s output. The market demand function is QD $=40050$P$,$ where P is the price per unit of output and QD is total market demand. $1\mathrm{.}1(4$ points$)$ Derive the market$$s direct supply function, QS $=$ f$($P$).$ Market direct supply function: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_1\mathrm{.}2(3$ points$)$ Determine the short$-$run equilibrium market price $($P$*)$ and total quantity $($Q$*).$ P$*=$ $ $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$ Q$*=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$ units.