Crowd Control
The local police department has recently been told that a major 4 th
of July event was going to take place in their city. In trying to put a plan in action that would accommodate visitors entering the city as well as commuters leaving the city, they collected data from another event at a city half their size.
Time of Day (hours) # of people per city block
4:00 pm 25
5:00 pm
6:00 pm
7:00 pm
8:00 pm
120
300
750
1100
9:00 pm
10:00 pm
11:00 pm
1325
1300
950
12:00 am
1:00 am
450
50
Given the additional police force that will be on duty during the celebration, the estimate they will be able to contain 400 to 600 people per block to the sidewalks.
After the crowds grow beyond that, they will be force to close the streets. They will not want to close the streets early due inconvenience it cause to the commuters.
1) Make a scatter plot of the projected event population the city will see based on the data above.
2) What parent function has the same approximate shape? What should the range and domain be for this specific problem and how would that be helpful in creating a graph?
3) Using a graphing calculator determine a function that models the new data. Does the calculator function represent the data? Could it be used as a predictor function?
4) Using the function from part 3, what time should the police close the roads, and again after the event reopen the roads? Explain how to calculate using your function in #3. How could these times be represented graphically?
Teacher Notes
Standard/s Addressed:
8.4
GLE - 1.6.2, 1.6.4
Scaffolding Questions:

Why would you use math in this circumstance rather than just setting an arbitrary time?

Sample Solutions:
1) Possible graph of data:
People per city block at time (t)
3000
2500
2000
1500
1000
Series1
500
0
0 2 4 6 8 10 12 14
Time (t) in Hours
2) Parent function looks similar to an upside-down parabola or of the function would be limited by y

0 y
  x
2
. The range
and less than the largest number of people who can physically fit in a city block. The domain would be the hours you would expect the crowds to start congregating and dispersing, ie.
 x : 3
 t

14

.
3) Using the calculator’s quadratic regression, the function would be f
 
 
117 .
5 t
2 
2090 t

7035
. This equation doesn’t quite fit the data. It does have similar x-intercepts and could be used to estimate time.
4) They would need to close the roads prior to the event when f
 
is between 400 and 600. This would happen at the earliest at 5:00 pm for 400, and the latest at
5:15. They would need to reopen the road at around 12:45 for 600 people and
1:00. for 400 people. To represent this graphically, you could draw horizontal lines at y=400 and y=600 to see the intercepts of the graph. These would be the times at which the roads would have to be shut down or opened.
Extension questions:
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