Question $3(25$ pts$.)$: The speed$-$density relationship of a road is known to be:

$u=u_f[1-(\frac{k}{k_j}{)}^{1\mathrm{.}5}]$

where $u$ is in MPH and $k$ is in veh$/$mi$.$ It is also known that the jam density $(k_j)$ is $150ve\frac{h}{m}$i and capacity $(q_{\text{c}\text{a}\text{p}})$ is $3664$ veh$/$hr$.$ Determine the following:

a$)$ Density at maximum flow $(k_{\text{c}\text{a}\text{p}}).$

b$)$ Speed at maximum flow $).$

c$)$ Free flow speed $(u_f).$

d$)$ Derive the necessary equation to solve for speeds at flow $=2000ve\frac{h}{h}r.$ You don't need to solve it$.$ Please specify which equation you use to solve for kcap as well.