# --- Hot Air Balloons ---

### By: Kritima Lamichhane & James McAbee

## 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.

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**Q.2**

** Here is a table of the values and a graph that show the first balloon’s altitude every 5 minutes (every 100 ft.) beginning at 5 minutes before the balloon was sighted at an altitude of 800 ft. until the balloon lands (an altitude of 900 ft. to an altitude of 0 ft).**

**This table shows the relationship between the the altitude in feet of the balloon, with the corresponding time. For example, 20 minutes after the balloon is sighted it is at 400 feet.**

**The title of this graph would be "Balloon Altitudes even Before it Was Sighted". Even before someone saw the balloon, it was still flying. It was at higher altitudes before it was sighted.**

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**Q.3**

**If the balloon has been dropping at a rate of 20 ft. per minute and 100 ft. every 5 minutes, then 5 minutes before the balloon was sighted at an altitude of 800 ft. the balloon was at an altitude of 900 ft. **

**The title of this graph would be "Balloon Altitudes even Before it Was Sighted". Even before someone saw the balloon, it was still flying. It was at higher altitudes before it was sighted.**

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**Q.4**

**If it takes the balloon 40 minutes to reach an altitude of 0 feet and the balloon is dropping at a rate of 20 ft. per minute then it would take 39 minutes for the balloon to reach an altitude of 20 ft.**

**It would take the balloon 40 minutes to land.**

**The title of this graph is "Balloon Sightings at a Certain Altitude". It shows the balloon's altitude at exactly 20 feet. When it does reach 20 feet, it has been 39 minutes, since the sighting of the balloon.**

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**Q.5**

**If a balloon is sighted at 1200 feet and is descending at 20 feet per minute the function would be f(x) = 1200 - 20 x. It takes 20 more minutes for balloon number 2 to land then balloon number 1. They both descend at twenty minutes per minute but balloon number 2 starts at 1200 feet while b**

**alloon number 1 was sighted at 800 feet.**

**This graph's title would be " A Balloon Sighting a Little Bit Higher". This graph represents the what the new balloon sighting (red) looks like compared to the first balloon (green). The new balloon starts at a higher altitude, so it takes a longer time to land then the first balloon does. That is why the red line is higher then the green line, and also why the red line goes farther on the x-value then the green line does. It goes farther because it starts higher and it lands later.**

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**Q.6**

**f(x) = 800 - 30 x -- function **

**It takes the third balloon much less time to reach ground level because it is decreasing it's altitude by 30 feet per minute, while the first balloon is decreasing it's altitude by 20 feet per minute. It takes the first ballon 40 minutes to land while it only takes the third balloon 26 2/3 minutes to land.**

## 1st and 3rd Balloon GraphThis graph shows the balloon sightings for both the first and third balloon sightings, together, to show the differences. Green: Third Balloon Red: First Balloon | ## 1st Balloon's Table This table shows the first balloon's altitude with 5 min time increments. | ## 3rd Balloon's Table This table shows the third balloon's altitude with 5 min time increments. |

## 1st and 3rd Balloon Graph

This graph shows the balloon sightings for both the first and third balloon sightings, together, to show the differences.

Green: Third Balloon

Red: First Balloon

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**Q.7**

**Function for the fourth balloon: f(x) = 30x**

**Function for when the first and fourth balloons are at the same altitude: **

**30(16) = 800 - (20 * 16)**

**The first and fourth balloons would be at the same altitude at 16 minutes. That altitude is 480 feet.**

**The title of this graph would be "FIrst and Fourth Balloons". This graph shows the first balloon's (Green) and the Fourth Balloon's (Red) altitude and the times they are at that altitude. It also shows their point of intersection, (16, 480).**

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**Q.8**

**If the first and the third balloons both started their descent at the same time, at what altitude would the third balloon have to start at to be able to land at the same time as the first ballon.**

**The third balloon would have to start at 1200 feet. The function or equation is:**

**800 - 20(40) = 1200 - 30(40).**

**This equation represents both equations and where they intersect. The equation shows that they both intersect each other at 40 minutes. The 40 minutes represents when they both land. This proves that starting the third balloon at 1200 feet makes it land at the same time as the first balloon. The title of the graph below would be "Balloons Landing at the Same Time".**