Hot Air Balloon Project!
By: Rachael Kang and Alice Han
A hot air balloon that was sighted at 800 feet is descending at 20 minutes per minute. The function relating to the variables would be f(x)=800-20x, x representing the number of minutes.
The balloon was 900 feet 5 minutes before it was sighted. For every 5 minutes the balloon descends it descends at a 100 feet, and the altitude the balloon was sighted at 800 feet. Therefore the altitude of the balloon was 5 minutes before it was sighted was 900 feet (800+100).
It also takes 5 extra minutes to land since it was sighted 5 minutes earlier. Therefore it lands at 45 minutes instead of 40.
The hot air balloon takes 39 minutes to reach an altitude of 20 feet. The graph below shows where the balloon would be in 39 minutes. The Y represents the altitude.
A second balloon is sighted at an altitude of 1200 feet and is descending at 20 feet per minute. The function that represents the situation would be f(x): 1200-20x, X representing the number of minutes. It takes 60 minutes to land while the first balloon took 40 minutes to land. The point where it would be the same altitude at the same time would be never, since they are going down at the same constant rate but at different starting altitudes, making them parallel to each other. (Look at second graph below)
This is the graph for the function, f(x)=1,200-20x. The 1,200 represents they Y intercept, and -20 is how much the balloon is going down per minute.
This is the relationship between the two functions. It doesn't cross each other, ever, because it's parallel.
The third balloon is first sighted at the altitude of 800 feet, but is descending at a faster rate, 30 feet per minute. The function that describes this situation would be f(x)=800-30x, X representing the number of minutes.
The third balloon takes less time to descend than the first. The third landed at 26 and two thirds of a minute and the first landed at 40 minutes. It has the same Y intercept which means it has the same altitude when it was first sighted, but the third balloon descended at a faster rate thus landing before the first. Use the graph below to see the relationship. (Disregard the blue line as that represents the second balloon.
The forth balloon is launched from the ground rising at the rate of 30 feet per minute. The function that would represent this situation is f(x)= 30x, X representing the number of minutes.
The balloons are at the same altitude at the same height, which is the intersection between the two lines on the graph below. The altitude is at 480 feet at 16 minutes.
The third balloon's altitude should be 1200 in order to reach the ground at the same time as the first balloon. The function equation that describes the situation would be f(x)= 1200-30x or Y= -30X+1200. Looking at the screenshot below, we first found the x intercepts for both balloons. The x intercept is when the line crosses the x axis. The first balloon's x intercept was 40, and the second was 26 and two thirds. Since we want to change when the third balloon lands, we canceled that out and multiplied 40 (the x intercept for the first balloon) and -30 for the rate of change for the third balloon. Then you have the slope equation of Y=-30(40)+b. Simplifying that will take us to y=-1200+b. Since we want 0 to be our Y (so it lands), we replace that with 0. 0=-1200+b. To find the B, we add 1,200 to both sides so we can have a variable on one side and the numbers on the other. Then we get 1,200=b, which means 1,200 is the answer!
Y=mx+b is the slope intercept form. M stands for the slope, or the change of rate. The B stands for y intercept, which is when the line crosses the y axis.