Traffic Modelling Conundrum?

📌 CONCEPT: The average number of people waiting at the bus stop can be calculated by integrating the rate at which people arrive over a given time period and then dividing by the total time period, excluding the last group that arrives right before the bus.

📐 RULE / FORMULA: The average number of people waiting is given by the formula: \[ \frac{1}{b-a} \\int_{a}^{b} f(t) \\, dt \] where f(t) is the rate at which people arrive, and a and b are the start and end times, respectively.

💡 WORKED EXAMPLE: To find the average number of people waiting over a period of 12 hours, excluding the last group that arrives right before the bus, we need to integrate the rate function f(t) = 2t^2 + 3t + 1 from a = 0 to b = 8 hours (since the bus arrives once every 4 hours) and divide by the total time period (8 hours).

⚠️ COMMON MISTAKE: Students often forget to exclude the last group that arrives right before the bus, which means they should not include the last interval of time in their integration.