Problem $9$

Consider the following $U(K)$ function, with $U$ in units of $m\frac{i}{h}r$ and K in units of $[$veh$/$mi$]$:

$U(K)=80-0\mathrm{.}001K^3,$

a$.(1$ pt$)$ What is $U_f,$ the free$-$flow speed represented by this equation? Present you answer to the nearest

$0\mathrm{.}1[$mph$].$

b$.(2$ pts$)$ What is $\frac{d^{2}U(K)}{d{K}^2}?$ Present numerical values to $4$ decimal places.

c$.(1$ pt$)$ Which of the following does the sign $($negative$,$ positive, zero$)$ of $\frac{d^{2}U(K)}{d{K}^2}$ best represent? Indicate one answer.

i$.$ Speed changes more when a few more vehicles "are added to the traffic" when the highway is congested than when the highway is near free flow

ii$.$ Speed changes more when a few more vehicles "are added to the traffic" when the highway is near free flow than when the highway is congested

iii. Speed changes the same amount when a few more vehicles "are added to the traffic" when the highway is near free flow than when the highway is congested

d$.(3$ pts$)$ For what values of density would you want to turn on a ramp meter?