Q$.3$ Given the speed$-$density relationship $($where $u$ is in mph and $k$ is in vehicles per mile$)$:

$u=0\mathrm{.}001**(k-250{)}^2-2$

Please calculate the following:

$($a$)$ Free$-$flow speed

$($b$)$ Jam density

$($c$)$ Speed and density at maximum flow

$($d$)$ Lane capacity