Optimal Flow Rate for Traffic

LaTex: $q(k) = v_{\text{max}} * k* (k_{\text{max}}-k)$ $q(k) = 100 {\text{ km/h}} * k *(200 {\text{ cars/km}}-k)$ $q(k) = 20000{\text{ cars/h}} * k - 100{\text{ km/h}}*k^2$ Now take the derivate of q(k) and you will get $q'(k) = 20000{\text{ cars/h}} * 1 - 100{\text{ km/h}}*2k$ For the maximum of k we will but q'(k) eqal to 0 $0 = 20000{\text{ cars/h}} - 200{\text{ km/h}}*k$ with some rearranging $200{\text{km/h}}*k = 20000{\text{ cars/h}}$ $k = 100{\text{ cars/km}}$ Now putting this in the original equation gives
$q(100) = 100{\text{ km/h}} * 100{\text{ cars/km}} *(200 {\text{ cars/km}}-100{\text{ cars/km}})$ $q(100) = 100{\text {km/h}} * 100{\text{ cars/km}} * 100$ $q(100) = \boxed{1000000{\text{ cars/h}}}$