# 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

f(x)=800-20x

x- the number of minutes balloon 1 has been descending

## Question 2

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

## Question 3

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

900 feet in the air.

## Question 4

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.

## 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.

## 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)

## 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.

## 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).