A loop detector has recorded the information shown in the figure. The loop width is $6$ feet. Each gate represents the brief period when a vehicle is in passing over the loop. $"$On$"$ means a vehicle is in the detection zone, while "off" means no vehicle is in the detection zone. The numbers above each gate represent the duration of each gate. For example, $"0-12"$ means a vehicle enters the detection zone at $\frac{0}{{60}^{th}}$ second and exits at $\frac{12}{{60}^{th}}$ second. The number of seconds is labeled at the bottom of the figure.

For example, $1,2,$ and $3$ mean the first, second, and third seconds, respectively. Assume the vehicle length is uniformly $15$ feet, and determine the following from the figure:

$($a$)$ Vehicle count during the observation period and the equivalent hourly flow rate

$($b$)$ Occupancy during the observation period

$($c$)$ Time$-$mean speed and space$-$mean speed

$($d$)$ Traffic density

$($e$)$ Estimate the speed from the speed$-$flow$-$density relationship and compare the result with that for $($c$).$ Is your estimated speed the time$-$mean speed or the space$-$mean speed?