Question

I'm Serena. For a science project, my friend Jack and I are launching three model rockets, one after another. We launch the first rocket, and then 3 seconds later, we launch the next one. And we're launching the final rocket three seconds after that, from a platform that is 20 feet high.

For our project, we need to predict the paths for all three rockets. We also need to estimate when they will all be in the air at the same time. [A graph that shows "Height of rocket (feet)" on the y-axis and "Time (seconds)" on the x-axis is shown. A red downturned parabola is shown and labeled "Path of the first rocket."]
We have calculated the path of the first rocket. It looks like this: a parabola that opens down. The y-axis is the height of the rocket in feet, and the x-axis is the time in seconds.
My friend Jack thinks we need to recalculate the graphs for the other two model rockets. But since the rockets are all the same, I think we can just shift the graph of the first rocket to find the graphs for the other two. [The graph is duplicated in green and shifts to the right, and then again in blue and shifts to the right and up. Then the rockets blast off again.]
What do you think? How can we use the graph of the first rocket to create the graphs of the second and third rockets? When will all three rockets be in the air at the same time?Evaluate the Conjectures:
2. Do you agree with Serena that you can draw the graphs for the other two rockets by shifting the functions? Or do you think that Jack is correct that you need to recalculate the other two? Explain. (2 points)
Analyzing the Data:
Suppose that the path of the first model rocket follows the equation
h(t) = −6 • (t − 3.7)2 + 82.14,
where t is the time in seconds (after the first rocket is launched), and h(t) is the height of each rocket, in feet.
Compare the equation with the graph of the function. Assume this graph is a transformation from f(t) = –6t2. What does the term –3.7 do to the rocket's graph? What does the value t = 3.7 represent in the science project? (What happens to the rocket?)
Again assuming a transformation from f(t) = –6t2, what does the term 82.14 do to the rocket's graph? What does the value h(t) = 82.14 represent in the science project? (What is happening to the rocket?) (2 points)
Serena and Jack launch the second rocket 3 seconds after the first one. How is the graph of the second rocket different from the graph of the first rocket? Describe in terms of the vertical and horizontal shift.
What is the equation of the second rocket?
They launch the third rocket 3 seconds after the second rocket and from a 20-foot-tall platform. What will the graph of the third rocket look like? Describe in terms of the vertical and horizontal shift.
What is the equation of the third rocket?
Answer the following questions about the three rockets. Refer to the graph of rocket heights and times shown above.
a. Approximately when is the third rocket launched?
b. Approximately when does the first rocket land?
c. What is the approximate interval during which all three rockets are in the air?

Answer 1

Regarding the conjecture of Serena and Jack:

Serena suggests that they can use the graph of the first rocket and shift it to find the graphs for the other two rockets. This means that the paths of the rockets are similar, and the only difference is the time of launch. Jack suggests that they need to recalculate the graphs for the other two rockets, which means that the paths of the rockets are different.

In this scenario, Serena is correct. Since the rockets are identical, they will follow the same path, but with a different time of launch. Thus, they can use the graph of the first rocket and shift it to the right to get the graph of the second rocket and shift it further to the right and up to get the graph of the third rocket.

Analyzing the equation:

The equation for the first rocket's path is h(t) = -6(t-3.7)^2 + 82.14. Assuming that the graph is a transformation from f(t) = -6t^2, the term -3.7 shifts the graph horizontally to the right by 3.7 seconds. This means that the first rocket was launched 3.7 seconds before the time t in the equation. The value t = 3.7 represents the time when the first rocket was launched.

The term 82.14 shifts the graph vertically up by 82.14 feet. This means that the initial height of the rocket is 82.14 feet above the ground. Therefore, the value h(t) = 82.14 represents the initial height of the rocket.

Equation of the second rocket:

The second rocket is launched 3 seconds after the first rocket. This means that the graph of the second rocket is a horizontal shift of the first rocket's graph by 3 seconds. Therefore, the equation of the second rocket is:

h(t) = -6(t-6.7)^2 + 82.14

This is because the launch time of the second rocket is t = 6.7 seconds (which is 3 seconds after the first rocket's launch).

Description of the third rocket's graph:

The third rocket is launched 3 seconds after the second rocket and from a 20-foot-tall platform. This means that the graph of the third rocket is a horizontal shift of the second rocket's graph by 3 seconds and a vertical shift upwards by 20 feet. Therefore, the equation of the third rocket is:

h(t) = -6(t-9.7)^2 + 102.14

This is because the launch time of the third rocket is t = 9.7 seconds (which is 3 seconds after the second rocket's launch).

Answers to the questions:

a. The third rocket is launched at approximately t = 9.7 seconds.

b. The first rocket lands when h(t) = 0. Solving -6(t-3.7)^2 + 82.14 = 0 gives t = 5.16 seconds (approximate).

c. The approximate interval during which all three rockets are in the air is from t = 6.7 seconds (when the second rocket is launched) to t = 14.46 seconds (when the first rocket lands).