Choose an Elevator
Based on
certain criteria,
determine which of two elevators reaches the lobby first.
Problem Statement
Two elevators leave the
nth floor at 2:00 P.M. The faster elevator takes one minute to travel
between floors and the slower elevator takes two minutes to travel 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 an 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.
Problem setup
To solve this problem, I will have to draw up a time
table for each elevator, showing the time it arrives and leaves on each
floor. This is similar to other pattern problems we have done in
class.
Plans to
Solve/Investigate the Problem
To find the solution to this problem, I will need to
track the times that each elevator would arrive and leave each floor.
Once I see which elevator arrives first, I will add 3 minutes to the time
for the departure time, as the first elevator on each floor has to be there
for 3 minutes to pick up passengers. The later elevator will
essentially get to arrive and leave within the same minute, as they do not
have to pick up passengers. If both elevators get to the same floor at the
same time, neither picks up passengers, and goes directly to the next
floor, just as if they were too late to pick up passengers.
Investigation/Exploration
of the Problem
I made a table to help me see what each elevator did.
I decided to start the elevator on the 9^{th} floor, and see
what happened. Both elevators start at 2:00 p.m. The slow elevator
takes 2 minutes to get to a new floor. The fast elevator takes 1
minute to get to a floor. Whichever elevator arrives first has to wait 3
minutes to take on passengers. If an elevator gets to a floor 2^{nd},
it can leave immediately to the next floor. If both arrive at the same time
to the same floor, both elevators skip the floor and go to the next floor.
Floor

Slow elevator – stop

(+3 minutes if first to
floor)

Slow elevator – leave

Fast elevator – stop

(+ 3 minutes if first to
floor)

Fast elevator  leave

9




2:00




2:00

8

2:02


2:02

2:01

+3

2:04

7

2:04

+3

2:07

2:05


2:05

6

2:09


2:09

2:06

+3

2:09

5

2:11


2:11

2:10

+3

2:13

4

2:13

+3

2:16

2:14


2:14

3

2:18


2:18

2:15

+3

2:18

2

2:20


2:20

2:19

+3

2:22

1

2:22**



2:23



I decided to carry it out for one more row, so I
started the elevators on the 10^{th} floor.
Floor

Slow elevator – stop

(+3 minutes if first to
floor)

Slow elevator – leave

Fast elevator – stop

(+ 3 minutes if first to
floor)

Fast elevator  leave

10




2:00




2:00

9

2:02


2:02

2:01

+3

2:04

8

2:04

+3

2:07

2:05


2:05

7

2:09


2:09

2:06

+3

2:09

6

2:11


2:11

2:10

+3

2:13

5

2:13

+3

2:16

2:14


2:14

4

2:18


2:18

2:15

+3

2:18

3

2:20


2:20

2:19

+3

2:22

2

2:22

+3

2:25

2:23


2:23

1

2:27



2:24**



By this, I deducted that if “n”
was 9 floors, then by the
slower elevator would beat the faster elevator. If “n”
was 10 floors, then the faster elevator would beat the slower one.
Extensions of the Problem
Inn
looking at the tables above, it seems that every 3 floors, the slower
elevator is first. Likewise, the faster elevator will be first to the
floor 2 out of every 3 times. It would be interesting to change this
problem up a little bit by altering how long the first elevator must wait
for passengers. I wonder if the time for passengers was increased
from 3 minutes to 4 minutes, what would the pattern be? What would the pattern be if it was
decreased to 2 minutes wait for passengers?
Author & Contact
Meg Ramsey
mramsey@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
Math.com
BJPinchbeck
U
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