Accumulated Effect of a Rate Part 2
Let's discuss one intuitive way of answering the question posed in the previous activity. You might say to yourself, "I'll use the model to get a couple traffic rates between 6am and 4pm, and then multiply these rates with a period of time to estimate car count in the period, and then add up the estimates to get a total traffic flow estimate."
This is reasonable since a rate of "cars per minute" times "minutes" will indeed give a car count.
Let's explore the arithmetic behind how this would play out to estimate the traffic between 6am and 4pm. Bear with me. It's a little intricate, but it's only arithmetic.
- There's a total of 10 hours between 6am (minute 360) and 4pm (minute 960), so maybe we'll split the interval into two periods: "morning" from 6am-11am, and "midday" from 11am to 4pm.
- Next, we'll obtain two rates, one at 6am, and another halfway through the day at 11am (minute 660) to serve as the morning and midday traffic rates. These rates are
g(360)=3.38769cars per minute in the morning andg(660)=9.47717cars per minute for midday. Keep in mind: we are only trying to get an estimate of car count through the day, so it's ok that we use one rate for the entire morning, and one rate for the entire midday. - Now, let's use these rates to estimate the morning car count and the midday car count by multiplying these rates times the number of minutes in each period. In the morning, this would work out to 3.38769 cars per minute times 300 minutes (5 hours), which is 1016.307 cars. In the afternoon, we'd have 9.47717 cars per minute times 300 minutes, which is 2843.151 cars.
- Finally, we'd add the morning and midday car count estimates, round to sum to the nearest car, and obtain an estimate of 3859 total cars between 6am and 4pm.
g(x) output in "cars-per-minute" multiplied with a number of "minutes" equals cars. We're calling it a "car count", but that's it.
In fact, I'd contend that this a pretty intuitive method, and I bet you'd have maybe even come up with it by yourself. Working out the arithmetic makes it seem more complicated than it actually is. So let's take most of the arithmetic out to try to really see the core ideas more clearly:
- Break 6am to 4pm into 2 five hour periods: "morning" from 6-11 and "midday" from 11-4.
- Use the traffic rate at 6am and at 11am for estimates of the traffic rate throughout the entire morning and the entire midday, respectively. (We totally understand this will not be perfectly accurate, but that's ok for now)
- Use these rates to get morning and midday car count estimates
- Add the car count estimates together to get a total car count estimate for the day from 6am to 4pm.
g(360)=3.38769 cars per minute, and base of 300 minutes, and so has area exactly equal to 1016.307 cars. Similarly the area of the right rectangle is 9.47717*300=2843.151 cars. The TotalTraffic is the sum of the areas of these two rectangles. DO NOT RUSH HERE. This is an absolutely critical conceptual connection to make on the path to the integral
Now I ask you: how can you improve this method to get a better estimate of the total traffic between 6am and 4pm?
There are a lot of different things you might try, but one improvement is "extensible", which means that it can be systematically applied to all sorts of other mathematical models, and it is also the idea that leads to the integral. Move to the next activity to check it out.