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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 9th 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 2nd, 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 10th 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

 

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