$ALGEBRA WORD PROBLEM - Trajectory of a NASA rocket$

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$ALGEBRA WORD PROBLEM - Trajectory of a NASA rocket$

NASA recently tested a rocket that was launched at their headquarters to find the direction that it would take and how long it would be able to move upwards before making a turn and descend back to the ground. after being tested, the group of scientists carrying out the experiment noticed that the rocket reached its maximum height 5 minutes after being launched. The height that the rocket attains over time is modeled by the following function $F(t)= -2t^2+20t+40$ where $t$ represents the time elapsed in minutes and $F(t)$ is the height attained by the rocket in meters over time.

What is the height of the rocket at 2 minutes, at 3 minutes and at 8?
What is the maximum height that the rocket attains before turning back to the ground?
How long does it take for the rocket to hit the ground as it turns around from it's trajectory?
Use a coordinate system that includes both units of time and height to describe the trajectory of the rocket? label all the info asked to in prev questions.
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a) The height of the function is given by the function $F(t)= -2t^2+20t+40$. So the height of the rocket at 2 minutes, at 3 minutes and at 8 is

\[F(2)= -2 (2)^2+20(2)+40=72,\]

\[F(3)=-2 (3)^2+20(3)+40=82,\]
\[F(8)=-2 (8)^2+20(8)+40=72.\]
b) We have
\[F(t)= -2t^2+20t+40=-2(t^2-10t-20)=-2(t^2-10t+25-45)\]
\[=-2((t-5)^2-45)=-2(t-5)^2+90.\]
Hence
\[F(t)=-2(t-5)^2+90.\]
Hence the the maximum height that the rocket attains before turning back to the ground is
\[F(5)=90.\]
c) How long does it take for the rocket to hit the ground as it turns around from it's trajectory?
We have
\[F(t)=-2(t-5)^2+90=0\]
\[\Rightarrow 2(t-5)^2=90 \Rightarrow (t-5)^2=45 \Rightarrow (t-5)=\pm \sqrt{45}\]
\[\Rightarrow t=5\pm \sqrt{45}.\]
Since $5- \sqrt{45}<0$ it is not acceptable. So the rocket to hit the ground at
\[t=5+ \sqrt{45}.\]

d) See the attached file.

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$ALGEBRA WORD PROBLEM - Trajectory of a NASA rocket$

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