Hot-Air Balloon

Griffin H, Damon C-S, Marlon S

Question #1

Our group had gone to the West Texas Balloon Festival for the day. While we were there, we spotted a blue balloon 800 feet in the sky. While we were watching we noticed that it was coming down at a rate of 20 feet per minute {F(x)=800-20x}. The amount of time that had passed after we spotted the balloon is the independent variable and the altitude of the balloon is the dependent variable because the amount of time that had passed determines what altitude the balloon was at.

Question #2

We took notice of the balloon coming down and recorded the height of the blue balloon every five minutes into a table and a graph. A pattern in the table and the graph's straight line showed that the balloon descended constantly at 100 feet every 5 minutes. 40 minutes after the balloon was sighted, it finally reached the ground (or an altitude of 0 feet).

Question #3

We wanted to know how high the blue balloon was 5 minutes before we spotted it. Since we knew from the table that the balloon was descending at 100 feet every 5 minutes, we simply added 100 feet to the altitude the balloon was at when we first sighted it (800 feet). It turned out that the altitude of the balloon was 900 feet 5 minutes before we sighted it.

Question #4

We then wanted to know haw long the blue balloon would take for the balloon to reach an altitude of 20 feet. Looking from the table and graph, we determined that it would take 39 minutes because since it's descending at a rate of 20 feet per minute and it would take 40 minutes for it to land, descending to an altitude of 20 feet would mean that the balloon is 1 minute away from touching the ground.

Question #5

Later during the festival, we spotted a red balloon at a height of 1200 feet and it appeared to be descending at 20 feet per minute {f(x)=1200-20x}. And when we recorded this function in a graph, we could tell by the x-intercept of the line that the red balloon would land 60 minutes after we spotted it; which is 20 minutes longer than the blue balloon's descent. When we compared the the line of the red balloon to the line of the blue balloon on the graph, we noticed that the lines were parallel to each other, meaning that they're descending at the same rate (20 feet per minute).

Question #6

Shortly, we spotted a green balloon at an altitude of 800 ft that was descending at a rate of 30 feet per minute {f(x)=800-30x)}. When we graphed the function, we could tell from the x-intercept of the line that the green balloon will land 26 2/3 minutes after we sighted it; which is 13 1/3 minutes less than the blue balloon's descent. When we compared the line of the green balloon to the line of the blue balloon on the graph, we saw that both of the lines had the same y-intercept (800); which means that the blue balloon and the green balloon were sighted at the same altitude; which was 800 feet.

Question #7

At the same time we first spotted the blue balloon, we also sighted a pink balloon that was rising up in the air at what appeared to be a rate of 30 feet per minute {f(x)=30x}. When the pink balloon reached 16 minutes of ascent, we saw that it was aligned with the blue balloon at an altitude of 480 feet. When we graphed the pink balloon's line and the blue balloon's line together, we found that the intersection of the lines (16, 480) represented when the two balloons would align with each other and the altitude that it will occur at.


Question #8

We soon wanted to figure out the altitude the green balloon would have to be at when we saw it to reach the ground at the same time as the blue balloon. We figured this out by using the function {f(x)=30x} from x being equal to 0 to x being equal to 40. We saw that the green balloon would ascend to 1,200 feet in 40 minutes. We then concluded that the green balloon would have to be sighted at an altitude of 1,200 feet in order to land at the same time as the blue balloon (40 minutes). The new function for the line of the green balloon (colored black) on the graph is {f(x)=1,200-30x}.