$22.$ The demand equation for a quantity $\backslash ($ q $\backslash )$ of a product at price $\backslash ($ p $\backslash ),$ in dollars, is $\backslash ($ p$=-5$ q$+4000\backslash ).$ Companies producing the product report the cost, $\backslash ($ C $\backslash ),$ in dollars, to produce a quantity $\backslash ($ q $\backslash )$ is $\backslash ($ C$=6$ q$+5\backslash )$ dollars.

$($a$)$ Express a company's profit, in dollars, as a function of $\backslash ($ q $\backslash ).$

$($b$)$ What production level earns the company the largest profit?

$($c$)$ What is the largest profit possible?