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.


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