# Hot Air Balloons

## Introduction

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.

## Tables

These tables start 5 minutes before the balloon was sighted and show the balloon's altitude every 5 minutes until it reaches the ground.

This graph below shows the altitude of the balloon 5 minutes before it was sighted. Five minutes before would be -5 on the graph. The corresponding y-value would be 900 for an x-value of -5.

The graph below shows that it takes 39 minutes to reach an altitude of 20 feet. Using the

g-solve feature on the calculator we can tell the calculator what the # of feet it is (y) and it will tell us the #of minutes (x).

It will take 40 minutes for the the balloon to reach 0 feet (land). By looking at the graph, when the y value is 0 (feet) The x value is 40 (minutes).

The graph below shows how long it takes for the first balloon to land. It also shows how long it takes the second balloon (which starts at an altitude of 1200 feet and descends at 20 feet per minute) to land. The function for the second balloon is f(x)= 1200-20x. The first balloon is in blue, and the second balloon is in red. Based on the x intercepts, it takes 20 minutes more than the first balloon for the second balloon to land. The rate of descent (slope) is the same, but the second balloon started 400 feet above the first. This means that the lines are parallel.
The function for the third balloon is f(x)= 800-30x which means that this third balloon starts at an altitude of 800 feet and descends at a steady rate of 30 feet. According to the x-intersepts it takes the third balloon minutes and 20 seconds less than the first balloon to reach the ground. The third balloon descends at a steady rate of 30 feet per minute. The slope for the third balloon is steeper than the first balloon's slope. This shows that the third balloon reaches the ground 13 minutes and 20 seconds before the first balloon.

f(x)=30x The 1st and 4th balloons would be at the same altitude in 16 seconds. They will be at 480 ft in 16 seconds.