Problem $1$ on speed$-$flow$-$density diagram $(24$ points$)$:

A traffic engineering study has revealed the following attributes for a specific highway segment:

Free$-$flow speed $V_f\mathrm{.}80mph$

Maximum roadway capacity $(q_{\text{c}\text{a}\text{p}})$:$2,400$ vehicles per hour $(vph)$

With these values given, perform the following task:

Derive the jam density and the speed$-$density $($V$-$K$)$ relationship based on the Greenshields linear model,

Using the $V-K$ relationship, graphically represent the flow$-$density and speed$-$flow characteristics.

On the flow$-$density diagram, label the points representing the capacity, and jam density.

On the speed$-$flow diagram, label the free$-$flow speed, and the speed at capacity.