Hot Air Balloon Math Project

By: Anjali Massand and Jessica Krampitz

Problem Question

At the West Texas Balloon Festival, a hot air 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.

Function

The function that relates these variables and best describes this situation is:
f(x)= 800-20x

*X Axis and Y Axis

The X axis for all of the graphs is time (minutes) and the Y axis for all of the graphs is the altitude (feet).

Table of Balloon's Altitude every 5 minutes beginning at 5 minutes before the balloon was sighted until the balloon lands.

How high was the balloon 5 minutes before it was sighted? Show graph and explain.

The balloon was at 900 feet 5 minutes before it was sighted, because according to the graph above, if you start at 800 ft, then you go up or down by 100 feet every five minutes. So in this case, you would go up 100 feet from 800 feet, which is 900 feet.

How long does it take for the balloon to reach an altitude of 20 feet? How long does it take the balloon to land? Show graph and explain.

Whenever the altitude (y axis) is 20 feet, the x axis is 39 minutes. Therefore, it takes 39 minutes for the balloon to reach an altitude of 20 feet. It takes 40 minutes to land because both the x axis line and the graph line meet at 40 minutes. The first graph on this page shows the lines connecting at 40 minutes and the balloon landing, and the graph to the right shows the time is takes the balloon to reach an altitude of 20 feet (39 minutes).

2nd Problem Question

A second balloon is first sighted at an altitude of 1200 feet but is descending at 20 feet per minute. Write the function that represents this situation. How much longer does it take for the second balloon to land compared to that of the first balloon? How does the descent of the balloons compare? Show the graph. Explain the relationships between the two lines.

Function

The function that represents this situation is:
f(x)= 1200-20x

Graph for this Function (With both Balloon 1 and 2)

Balloon 1 has a red line with little green dots, and balloon two has a black line and big green dots.

How much longer does it take for the second balloon to land compare to that of the first balloon?

It takes the second balloon 20 more minutes to land compared to the first balloon, according to the graph above.

How does the descent of the balloons compare? Show the graph. Explain the relationships between the two lines.

They are both descending at a negative correlation and both going down by the same increments (amounts). They are also at about the same angle and seem to be parallel to each other. The lines just started their descent at different spots.You can tell those things from the graph above, with both balloon one and balloon two on there.

3rd Problem Question

A third balloon is first sighted at an altitude if 800 feet but is descending at 30 feet per minute. Write the function that represents this situation. How much longer does it take for the third balloon to land compared with that if the first balloon? How does the descent of the balloons compare? Show the graph. Explain the relationship between the two lines?

Function

The function that represents this situation is:
f(x)= 800-30x

Graph for this Function (With Balloon's 1, 2, and 3)

Balloon one has a red line with little green dots, balloon two has a black line and green dots, and the third balloon has a red line with big black dots.

How much longer does it take for the third balloon to land compared with that if the first balloon?

It takes the third balloon only 26.7 minutes to land, so it is at a faster descent than the first balloon, which took 40 minutes to land. We could tell the landing points from the graph above.

How does the descent of the balloons compare? Show the graph. Explain the relationship between the two lines?

The third balloon is steeper than the other two balloons, and is descending at a higher increment (faster). Also the third and first balloon are intersecting at 800 feet altitude. The second balloon is descending at a higher point than the other two balloons. All three balloons are still descending at a negative correlation.

4th Problem Question

At the instant the first balloon is sighted, the fourth balloon is launched from the ground rising at a rate of 30 feet per minute. Write the function that represents this situation. When will the first and fourth balloon be at the same altitude? What is that altitude? Show the graph. What does this mean graphically?

Function

The function that represents this situation is:

f(x)= 30x

Graph for this Function (With Balloon's 1, 2, 3, and 4)

*the colors of the lines of the different balloons changed. Balloon 1 is a red line with green dots, balloon 2 is an orange line with red dots, the balloon 3 is a green line with orange dots, the balloon 4 is a positive correlation with a green line and red dots.

When will the first and fourth balloon be at the same altitude? What is that altitude?

Yes they will be at the same altitude at about 480 feet, according to the graph right below.

Show the graph. What does this mean graphically?

This means that at about 480 feet the first and fourth balloon lines intersect. This means that they now have the same altitude when they intersect, as well. We figured that out from the graph to the right. The intersection of balloon 1 and 4 is circled in red.

At what altitude would the 3rd balloon have to begin its descent in order to reach the ground at the same time as the 1st balloon? What is the equation of the line?

The third balloon would have to begin it's descent at a 1200 feet altitude, in order to reach the ground and land at the same time as the first balloon (40 minutes). We set up this equation to figure that out:

800-20(40) = x-30(40)

800-800 = x- 1200

0 = x-1200

1200 = x


That would be the equation of this new third balloon line.