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Answer :
The quadratic function that models the height of the rocket is h(t) = at^2 - 6at + 9a + 20
The height of the rocket can be modeled by a quadratic function of time.
Let's define the time elapsed since the rocket was launched as t (in seconds) and the height of the rocket above the ground as h (in meters).
Since the height of the rocket is a quadratic function of time, we can express it as:
h(t) = at^2 + bt + c
where a, b, and c are constants that we need to determine.
Given that the rocket reaches its highest point at 20 meters above the ground 3 seconds after it was launched, we can use this information to find the values of the constants.
First, let's consider the vertex form of a quadratic function:
h(t) = a(t - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Since the rocket reaches its highest point, the vertex is at (3, 20). Substituting these values into the vertex form, we have:
h(t) = a(t - 3)^2 + 20
Expanding the squared term:
h(t) = a(t^2 - 6t + 9) + 20
Simplifying:
h(t) = at^2 - 6at + 9a + 20
Comparing this equation with the original quadratic function h(t) = at^2 + bt + c, we can determine the values of the constants:
a = a
b = -6a
c = 9a + 20
Therefore, the quadratic function that models the height of the rocket is:
h(t) = at^2 - 6at + 9a + 20
Since the specific value of a is not given, we cannot determine the exact quadratic function that represents the rocket's height. The value of a will depend on various factors, such as the initial velocity and acceleration of the rocket.
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