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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%.





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 the distance with the second step and each consecutive step is 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.


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