Optimal Flow Rate for Traffic

If the flow rate of traffic qq as a function of density kk is given by

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

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

Assume that vmaxv_{\text{max}} is 100 km/hr100 \text{ km/hr} and kmaxk_{max} is 200 cars/km200 \text{ cars/km}.

Note: 100 km/hr=60 mi/hr100 \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 cars/km200\text{ cars/km} in bumper-to-bumper traffic.

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