Choose an Elevator
Problem Statement
Two elevators leave the nth floor at 2:00 P.M. The faster elevator takes
one minute between floors and the slower elevator takes two minutes between
floors. The first elevator to reach a floor must stop for three minutes to take
on passengers. If both elevators arrive at a floor at the exact same time, they
become confused and do not stop for passengers.
If the final stop for the elevator is the lobby (1st floor), then describe n
if the faster elevator arrives at the lobby first. Describe n if the
slower elevator arrives at the lobby first.
(Source: Adapted from Mathematics Teaching in the Middle School, Apr 1994).
Problem setup
Here, I am not finding out how many
floors are in this building. I am being asked to determine a pattern for
the floor total if the faster elevator comes in first or if the slower elevator
comes in first.
Plans to Solve/Investigate the
Problem
Initially, I used pencil and paper to
write down the possible arrival times. This proved to be rather arduous,
and I returned to my first love...Excel!
Investigation/Exploration of the
Problem
After much time and effort, I
realized that I had overlooked an important part of the problem. The
first
elevator to reach a floor must stop for three minutes to take on passengers.
Below is a part of the initial and erroneous spreadsheet:
Floor 
Faster 

Slower 
N 
0.083333 

0.083333 
N1 
0.084028 
1 
0.084722 
N2 
0.086806 

0.086111 
N3 
0.0875 
1 
0.089583 
N4 
0.090278 
1 
0.090972 
N5 
0.093056 
1 
0.094444 
N6 
0.095833 

0.095833 
N7 
0.096528 
1 
0.099306 
N8 
0.099306 
1 
0.102778 
N9 
0.102083 
1 
0.10625 
N10 
0.104861 
1 
0.109722 
N11 
0.107639 
1 
0.113194 
N12 
0.110417 
1 
0.116667 
N13 
0.113194 
1 
0.118056 
N14 
0.115972 
1 
0.119444 
N15 
0.11875 
1 
0.122917 
N16 
0.119444 
1 
0.126389 
N17 
0.120139 
1 
0.129861 
N18 
0.122917 
1 
0.133333 
N19 
0.125694 
1 
0.136806 
Here, I had not taken into account
the first elevator situation. I proceeded to insert an if statement that
checks to see which time was lowest in an effort to determine which elevator
reached each floor first. The floor with the earlier arrival has a 1 to
the right of its time.
Please note that the time was
changed to a value using a formula in excel.
This determination told me where to
add the time for travel between floors only and where to include time for
passenger pick up and drop off. The resultant and hopefully correct
spreadsheet sample is below.
Floor 
Faster 

Slower 

N 
0.083333 

0.083333 

N1 
0.084028 
1 
0.084722 

N2 
0.086806 

0.086111 
1 
N3 
0.0875 
1 
0.089583 

N4 
0.090278 
1 
0.090972 

N5 
0.093056 

0.092361 
1 
N6 
0.09375 
1 
0.095833 

N7 
0.096528 
1 
0.097222 

N8 
0.099306 

0.098611 
1 
The floors where the slower cab
arrives first are highlighted in green (the color of the season I might add).
Now the ultimate question was what is
true about the number of floors in the building if the slower car reaches the
lobby first. Take a look at each place where the floors are shaded green.
If these places were the lobby of the building, then the place would have 3, 6,
9, 12,... floors (a multiple of 3). If the faster cab reaches the lobby
first, the number of floors could be anything EXCEPT a multiple of 3.
Below is a link to the entire
unsophisticated spreadsheet...
Elevators
Extensions of the Problem
What would the case be if there were
3 elevators, the middle one takes 1.5 minutes to travel between floors?
Author & Contact
Michelle Houston
Link(s) to resources, references, lesson plans, and/or other
materials
