A toy rocket is placed on a tower and launched straight up. The table shows its height, y, in meters above the ground after x seconds.
time, x (s) 0, 1, 2, 3, 4, 5, 6, 7, 8
height, y (m) 16, 49, 72, 85, 88, 81, 64, 37, 0
a) Does the data in the table represent a quadratic relation? Explain.
b)Use the regression method to determine the equation of your curve of good fit.
c) Show the graph and what is the maximum height of the rocket? (label that point on the graph)
d) What is the height of the tower?
e) How long is the rocket in flight?
a) The data in the table represents a quadratic relation because the height of the rocket changes according to a consistent pattern. As the time increase, the height initially increases, reaches a peak, and then begins to decrease.
b) Using regression, we can determine the equation of the curve of good fit. By analyzing the data, we can determine that the quadratic equation takes the form y = ax^2 + bx + c.
Using the given data, we can have the following system of equations:
a(0)^2 + b(0) + c = 16
a(1)^2 + b(1) + c = 49
a(2)^2 + b(2) + c = 72
a(3)^2 + b(3) + c = 85
a(4)^2 + b(4) + c = 88
a(5)^2 + b(5) + c = 81
a(6)^2 + b(6) + c = 64
a(7)^2 + b(7) + c = 37
a(8)^2 + b(8) + c = 0
By solving this system of equations, we can determine the values of a, b, and c.
c) In order to graph the quadratic relation, we plot the time, x, on the x-axis and the height, y, on the y-axis. From the graph, we can determine the maximum height of the rocket by finding the highest point on the graph.
d) The height of the tower can be determined by finding the initial height of the rocket at time x = 0.
e) The rocket is in flight for the duration of time when the height, y, is above 0. Thus, we can determine the length of the rocket's flight by finding the time when the rocket reaches the ground (when y = 0).