Hot-Air Balloon

By: Abishek Balakrishnan and Zubair Siddiqui (Period 6)

Question 1

The function is f(x) = 800-20x, where 800 is the altitude when the hot-air balloon was first spotted, -20x is a decrease of 20 times the feet per minute (feet per minute is x, -20 is decrease in altitude), and f(x) is the altitude of the balloon at different times after it was first spotted.

Question 2

This question asks to graph and create a table for this function. In the function f(x) = 800-20x, f(x) is the y coordinate, x the x coordinate, -20 the slope (how much the graph decreases by, in this case, every time we go -20 up or 20 down, we go one to the right), and 800 the y intercept (where the graph touches the y axis).
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Question 3

In this question, we are trying to find the altitude during a certain time, f(x). We know that we must include 800 (altitude when balloon was spotted), minus 20 times the number of minutes after the balloon was spotted. This gives us f(x) = 800-20x. Now, we must plug in the numbers from the question. It gives us the number 5, but says this was 5 minutes before the balloon was spotted, and this can be rewritten as -5 minutes after the balloon was spotted. Then we do the work:

1.) f(x)= 800-20(-5)

2.) f(x)= 800- (-100), which is the same as f(x) = 800+100

3.) f(x)= 900


This gives us the answer, 900 feet.

Question 4

In this question we are trying to figure out how long it takes to reach a certain altitude. This means that the altitude of the balloon during this unknown period of time after the balloon had been spotted is given. This altitude is the f(x), and, in this case, is 20 feet (see question 1 for more on what f(x) represents in this function). Now we have our function: 20 = 800 (remember, this is always the same because it is the altitude when the balloon was first spotted, which doesn't change) minus 20x. Next, we do the work:

1). 20= 800-20x

2.) Subtract 800 from both sides to get, -780=-20x

3.) Now divide both sides by -20 (to isolate the x), and get 39= x

4.) Rewrite 39=x, and get x= 39 (number=x can always be rewritten as x=number)


This leads us to the answer, 39 minutes.

Question 5a

In this question, we are doing two things, writing a new function for a different balloon, and comparing it to the first. As stated earlier, in the function f(x)= 800-20x, f(x) is the altitude at a certain time after the balloon was spotted, 800 is the altitude it was first spotted at, and -20x is a decrease of 20 times the feet per minute. Using this, we can create a new function. This second balloon was first spotted at 1200 feet (altitude), so this is the equivalent of the 800 in the previous equation. The second balloon is descending at the same rate as the first one, so it is also -20x. This gives us the function f(x) = 1200-20x.

Question 5b

However, this is only the first part of the question. Now we have to see how much longer it will take for the second balloon to land than the first one. Landing tells us f(x) will be 0 (see Question 1 for why). We just have to plug this into both functions, and find the difference. For the first function:

1.) 0=800-20x

2.) Subtract 800 from both sides, and get -800=-20x

3.) Divide both sides by -20 (to isolate x), and get 40=x

4.) 40 minutes


For the second function:

1.) 0=1200-20x

2.) Subtract 1200 from both sides, and get -1200=-20x

3.) Divide both sides by -20, and get 60=x

4.) 60 minutes


Now we subtract to see how much longer it took for the second balloon to land.

1.) 60-40=20. It took 20 more minutes for the second balloon to land!

Question 6

Now we are writing another function, this time for a third balloon. It was first spotted at an altitude of 800 feet, just like the first one, so that part of the first function will stay the same. However, this balloon is decreasing faster than before, 30 times the feet per minute! So, if we follow the structure of the first function, we will end up with f(x) = 800-30x. Now, using what we did in Question 5b, we must find how much longer it'll take for the third balloon to land than the first one. We already have the answer to the first function (40 minutes to land), so we only need to calculate how long it takes to land for the third one.

1.) 0=800-30x

2.) Subtract 800 from both sides to get -800=-30x

3.) Divide both sides by -30, and get 26 2/3=x

4.) 26 minutes and 40 seconds (same thing as mixed fraction above)


Next, we do 26 2/3 -40=-13 1/3. It took 13 minutes and 20 seconds fewer minutes for the third balloon to land than the first one (negative more minutes is the same as positive less minutes)!

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Question 7

In this question, we are creating yet another function, this time for a fourth balloon. The question doesn't tell the fourth balloon's altitude when it was spotted, so we leave this out of the function. However, it does tell you that the balloon is rising 30 feet per minute. This is different than the prior functions, because those decreased at a certain rate. This means that instead of a negative number being multiplied by x, it will be a positive one. This means that the function is f(x) = 30x. Next, the question is asking for when the first and fourth balloons will be at the same altitude. To show this, we make the two functions equal each other. Since both functions equal f(x), we can leave this out, and just write the rest. This would be 800-20x=30x. This is how we solve this equation:

1.) 800-20x=30x

2.) Subtract -20x from both sides (same as adding 20x to both sides) to get the x's all on one side. This will get you 800=50x

3.) Divide both sides by 50, and get 16=x


This means that both balloons would have the same altitude after 16 minutes of the first balloon being spotted and the fourth one being launched. To find the altitude, we would just plug 16 in place of x. It could be either y=30(16), or y=800-20(16), which equals 480 feet. (y is the altitude). The equations would equal the same thing because they intersect after 16 minutes.

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Question 8

In this question, we are trying to calculate how high the third balloon should be if we want it to land at the same time as the first balloon. We can use a simple equation to help us find this out. y=mx+b. This equation is called the slope intercept equation. We have y, which is 0, because when they land the are both on the ground and not up in the air. We also have m and x. M is the feet per minute it falls, which is -30, and x is how long it takes to land, which is 40. Now we just plug the numbers in and we get out equation.

1.) We just substitute the numbers. 0=-30(40)+b

2.) We now solve the equation. -30 times 40 equals -1200. 0=-1200+b

3.) We finish the equation by adding 1200 to both side to get 1200=b

4.) We now found our answer. The third balloon would have to be 1200 feet in the air for it to land at the same time as the first balloon. Thus, the equation for this line would be y=-1200+b.

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