# Hot Air Balloon

## Scenario

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.

## Question 1

What function relating the varibles best describes this situation?

f(x)=800-20x

x- the number of minutes balloon 1 has been descending

## Question 2

Make a table of values and a graph to show the balloon’s altitude every 5 minutes beginning at 5 minutes before the balloon was sighted until the balloon lands.

X- the number of minutes the balloon has been descending

Y1- balloon 1's distance from the ground (feet)

The first balloon's color is blue. The Y-axis is the altitude (or the height) of the balloon. Every mark on the Y-axis is 100 feet. The X-axis is the number of minutes the balloon has been descending. Every mark on the X-axis is 5 minutes.

## Question 3

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

900 feet in the air.

The 1st balloon's color is blue. The Y-axis is the altitude (or the height) of the balloon. Every mark on the y axis is 100 feet. The X-axis is the number of minutes the balloon has been descending. Every mark in the x axis is 5 minutes. This graph represents the 1st balloon descending at a steady rate of 100 feet every 5 minutes (or 20 feet a minute), until it reaches the ground. So therefore, the balloon would be at the 900 feet mark in the air 5 minutes before it was sighted. (5 minutes=100 feet)

## Question 4

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

1) It takes the balloon 39 minutes to reach the 20 feet mark.

2) It takes a total of 40 minutes for the balloon to reach the ground. It lowers 20 feet every minute.

1st balloon's color is blue. This graph represents the 1st balloon descending at a steady rate of 100 feet per minute, so based on the graph and tables (question 2), it took the balloon 39 minutes to be a the 20 feet mark, and 40 minutes to land in the ground.

## Question 5

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

1) f(x)=1200-20x

x- The number of minutes since balloon 2 was sighted

2 -3) Both balloons are descending at the same rate, but balloon 2 is spotted at a higher altitude. If they both descend 100 feet every 5 minutes, and balloon 2 is spotted 400 feet higher than balloon 1, then you would multiply 5 (the number of minutes it takes to descend 100 feet) by 4 (the number of 100's in the distance from the ground when balloon 1 has reached the ground) and you get 20 minutes left until balloon 2 reaches the ground.

The 1st balloon's color is blue, the 2nd balloon's color is red. This graph is showing the relationship between the 1st balloon and the 2nd while they are descending at the same rate; 100 feet per minute. Also it is showing that the 2nd balloon was spotted 400 feet above the 1st.

## Question 6

A third balloon is first sighted at an altitude of 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 of the first balloon? How does the descent of the balloons compare? Show the graph. Explain the relationship between the two lines.

1) f(x)=800-30x

X- The number of minutes since balloon 3 was sighted

2) The third balloon descends at a rate of 30 feet per minute, so therefore, it will land in the ground at about 26 and 1/2 minutes (800÷30≈26.5)

The 3rd balloon's color is green. This graph is showing the relationship between the first and the third balloon. It is showing that the third balloon takes less time to reach the ground than the first. Even though they both start at the same altitude, the third balloon descends faster (30 feet per minute).

## Question 7

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. 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?

1) f(x)=30x

X- The number of minutes since balloon 4 was launched

2) The 1st and 4th balloons meet 16 minutes after the 1st balloon was sighted and the 4th ballon was launched, at an altitude of 480 feet in the air.

The 4th balloons color is purple. The 4th balloon is launched when the 1st balloon is sighted (0 on the Y) and increases at a rate of 30 feet per minute. Balloons 1 and 4 meet at 16 minutes in to balloon 1's descent and balloon 4's ascent. This is at an altitude of 480 feet.

## Question 8

1. 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?

1) Balloon 3: f(x) = 1200-30x

1200-30x = 800-20x

X- The number of minutes since balloons 1 and 3 were sighted

Balloon #3 would have to start at 1200 feet in the air to land exactly at the same time as balloon 1 (40 minutes).

The green represents balloon 3's new function. This graph shows that if balloon 3 is sighted at an altitude of 1200 feet, then balloons 1 and 3 will both land on the ground 40 minutes after they are sighted.