Let's solve the problem step by step for **queue dissipation time**, **maximum queue length**, and **total delay**. --- ## **Step 1: Calculate Net Queue Growth per Interval** | Interval | Arrival Rate | Departure Rate | Net Change (veh/hr) | Duration (hr) | |--------------------|--------------|----------------|---------------------|---------------| | 3:00 – 3:30 PM | 150 | 900 | +600 | .5 | | 3:30 – 4:00 PM | 900 | 120 | -300 | .5 | | 4:00 – 8:00 PM | 120 | 150 | -300 | 4 | - **Net Change = Arrival Rate - Departure Rate** - **Positive:** Queue builds up. - **Negative:** Queue dissipates. --- ## **Step 2: Compute Queue Length Over Time** ### **Interval 1: 3:00 – 3:30 PM** - Net Change: **+600 veh/hr** - In .5 hr: \[ \text{Queue Increase} = 600 \times .5 = 300~\text{vehicles} \] - **At 3:30 PM:** Queue = **300 vehicles** ### **Interval 2: 3:30 – 4:00 PM** - Net Change: **-300 veh/hr** - In .5 hr: \[ \text{Queue Decrease} = 300 \times .5 = 150~\text{vehicles} \] - **At 4:00 PM:** \[ \text{Queue} = 300 - 150 = 150~\text{vehicles} \] ### **Interval 3: 4:00 – 8:00 PM** - Net Change: **-300 veh/hr** - Initial Queue: **150 vehicles** - Time to dissipate: \[ \text{Time to clear} = \frac{150}{300} = .5~\text{hr} = 30~\text{min} \] - **Queue clears at 4:30 PM** --- ## **Step 3: Final Answers** ### **A. Maximum Queue Length** - Occurs at **3:30 PM** - **Maximum Queue:** **300 vehicles** ### **B. Queue Dissipation Time** - Queue is gone by **4:30 PM** - **Dissipation Time:** **4:30 PM** ### **C. Total Delay (Vehicle-Hours)** Total delay is the area under the queue vs. time curve (sum of vehicle-seconds/vehicle-hours spent waiting in queue): #### **Interval 1 (3:00–3:30 PM):** - Triangle: \[ \text{Area} = \frac{1}{2} \times .5~\text{hr} \times 300~\text{veh} = 75~\text{veh-hr} \] #### **Interval 2 (3:30–4:00 PM):** - Trapezoid from 300 to 150 vehicles: \[ \text{Area} = \frac{1}{2} \times (300 + 150) \times .5~\text{hr} = 112.5~\text{veh-hr} \] #### **Interval 3 (4:00–4:30 PM):** - Triangle from 150 to vehicles over .5 hr: \[ \text{Area} = \frac{1}{2} \times .5~\text{hr} \times 150~\text{veh} = 37.5~\text{veh-hr} \] #### **Sum:** \[ \text{Total Delay} = 75 + 112.5 + 37.5 = \boxed{225~\text{vehicle-hours}} \] --- ## **Summary Table** | Quantity | Value | |--------------------------|-------------------------| | **Max Queue Length** | 300 vehicles | | **Queue Dissipation Time** | 4:30 PM | | **Total Delay** | 225 vehicle-hours | --- ### **Explanation** - Queue **builds up** when arrivals > departures, **dissipates** when arrivals < departures. - **Total delay** is the sum of all vehicle-hours spent in the queue until it is cleared. --- Let me know if you need a diagram or further breakdown!