Have you ever wondered what it might feel like to float weightless in space? One way to try it out is to fly on a special aircraft that astronauts use to train for their trips to space. Both NASA and the Russian Space Agency have been flying these for years. The way this is accomplished is to fly to a high altitude, drop down to gain speed, and then start a large parabolic path up in the sky. For a time ranging from 10 to 20 seconds, along the top part of the parabolic flight, an environment simulating zero gravity is created within the plane. This effect can cause some nausea in the participants, giving rise to the name ‘Vomit Comet’, the plane used by NASA for zero-G parabolic training flights. Currently there is a private company that will sell you a zero-G ride, though it is a bit expensive.
This lab will have you take a look at the parabolic path to try to determine the maximum altitude the plane reaches. First, you will work with data given about the parabola to come up with a quadratic model for the flight. Then you will work to find the maximum value of the model. Now for the data:

Height of a Zero-G Flight t Seconds After Starting a Parabolic Flight Path
Time t in seconds
2
20
40
Height h in feet
23645
32015
33715

To find the quadratic model, you will be plugging the data into the model . The data points given are just like x and y values, where the x value is the time t in seconds and the y value is the altitude h in feet. Plug these into the model and you will get equations with a, b and c.

Part 1: Write your 3 by 3 system of equations for a, b, and c. Solve this system, then put your solutions to the system to form a quadratic model of the data.