Title
Determine
the height of a hot air balloon over time based on a recursive condition.
Problem Statement
A hot air balloon moves up 200 feet during its first minute of flight. Then
it will continue to rise each minute thereafter for a distance of 80% of
the distance traveled the previous minute. If you ignore air pressure, will
the balloon fly out into space (in other words, will it rise indefinitely)?
Explain why or why not.
Problem setup
The
problem states that the hot air balloon is in upward motion and will rise a
fraction of the previous distance in a unit of time. The balloon rises 200
feet in the initial minute and 80% of that distance each minute there
after.
Plans to Solve/Investigate the Problem
I
plan on using Microsoft excel to see if I can come up with a math formula
for the pattern.
Investigation/Exploration of the Problem
At
first I looked at the problem by hand calculating the distance for 3
minutes. I multiplied the distance for each minute by multiplying the
previous distance by 80%.
Time

Distance

0

0

1 minute

200 feet

2 minutes

160 feet

3 minutes

128 feet


Total distance = 488 feet

In
3 minutes the balloon would rise 488 feet. Using the calculation; 0.8 x
previous distance, I was ready to use excel. In using excel, I found
that the balloon would never stop moving.
Click Here to see a spreadsheet file for the
above table.
Extensions of the Problem
A person walks toward
the door, one foot with the first step and ½ the distance with the second
step and each consecutive step is ½ the distance of the last one, do they
eventually hit the door? Using an excel spread sheet, I find that yes, you
will hit the wall eventually when the sum of your steps equals the distance
you are away from the wall. If, for example, you started walking around a
building, with no final destination, you would, like the balloon, never
stop moving.
Click Here to see a spreadsheet file for the
extension problem.
Author & Contact
cgriswold@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
Link 1
Link 2
