# 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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