Optimal Flow Rate for Traffic

If the flow rate of traffic $q$ as a function of density $k$ is given by

$q(k) = v_{\text{max}} k (k_{\text{max}} - k)$,

what is the maximum flow rate possible (in cars/hr)?

Assume that $v_{\text{max}}$ is $100 \text{ km/hr}$ and $k_{max}$ is $200 \text{ cars/km}$.

Note: $100 \text{ km/hr} = 60\text{ mi/hr}$, which is a typical speed limit for a US highway, and midsize cars are approximately 5 m long, corresponding to $200\text{ cars/km}$ in bumper-to-bumper traffic.