Hot-Air Balloons

BY: JADE N

At the West Texas Balloon Festival, a hot-air balloon is sighted at an altitude of 800 feet and appears to be descending at a steady rate of 20 feet per minute. Spectators are wondering how the altitude of the balloon is changing as time passes. (Balloon #1)

The function Y= -20x+800 represents this situation.

Here is a table that shows Balloon #1's altitude every 5 minutes beginning 5 minutes before the balloon was sighted:

Here is a graph of Balloon #1 five minutes before it was sighted:

900 feet

The red circle on the graph shows the point of how high the balloon was 5 minutes before being sighted.


Here is a graph of how long it will take Balloon #1 to reach an altitude of 20 feet and how long it takes it to land:

Y=-20x+800= 39 minutes (to reach 20 feet)

40 minutes to land

The green circle on the graph represents where the hot-air balloon is at about 20 feet. The red circle on the graph shows when the hot-air balloon lands.


A second balloon is first sighted at an altitude of 1200 feet but is descending at 20 feet per minute. (Balloon #2)

This function represents this situation:

Y= 1200-20x


Here is a graph that shows how much longer it takes for Balloon #1 to land verses Balloon #2 and the relationship between the two balloons:

The red circles represent when the balloons have landed. The descent of the balloons are similar in that they both descend at 20 feet per minute. Balloon #1 reaches the ground after 40 minutes and Balloon#2 reaches the ground after 60 minutes.


A third balloon is first sighted at an altitude of 800 feet but is descending at 30 feet per minute. (Balloon #3)

This function represents this situation:

Y= 800-30x


Here is a graph that shows how much longer it takes Balloon #3 to land compared to Balloon #1 and how they compare:

  1. The red circles on the graph indicate when the balloons reached the ground. It takes Balloon#1 about 10 more minutes to reach the ground than Balloon #3. The rate of descent of Balloon#3 is faster by 10 feet per minute than Balloon #1.


At the instant the first balloon is sighted, a fourth balloon is launched from the ground rising at a rate of 30 feet per minute. (Balloon #4)

This function represents the situation:

Y1= -20x+800 Y4= 30x Y1=Y4 Function: 30x=-20x+800


Here is a graph that shows when Balloon #1 and Balloon #4 will be at the same altitude:

The red circle on the graph represents where Balloon #1 and Balloon #4 intersect. This intersection occurs after 16 minutes of Balloon #1 descending and Balloon #4 rising and the altitude is about 500 feet.


Spectators are wondering what altitude Balloon #3 has to begin its descent in order to reach the ground the same time as Balloon #1. Here is the answer:

Balloon #3 would have to start descending at 1,200 feet in the air.


Here is a graph to display the information:

Here is an equation that describes the situation

-20x+800=-30x+1200