Problem $3.$ A school zone is designated on a $0\mathrm{.}5$ mile stretch of roadway with traffic flow that follows the Greenshields model. The school zone imposes a speed limit of $20mph$ for a period of $30$ minutes before school starts in the morning and after school is out in the afternoon. Traffic engineers collect data when school is not in session, and find that the jam density and free$-$flow speed for the roadway are $284$ veh$/$mi and $63m\frac{i}{h}r,$ respectively. The roadway flow is $90\%$ of capacity and uncongested just before the school zone begins. Assume that vehicles already in the school zone when it begins will continue driving at their current speed, but any vehicles entering the school zone will obey the reduced speed limit of $20mph.$

Draw a time$-$space diagram indicating all of the shockwaves that are formed as a result of the school zone speed reduction. Label the speed, density, and flow in each region of the diagram, and compute the speed and Tirection of every shockwave.