Hello!
The answer for A is:
[tex]X1=-1\\X2=5[/tex]
The answer for B is: The rocket will hit the lake surface in 5 seconds.
Why?
Defining the terms:
[tex]a=-16\\b=64\\c=80\\[/tex]
So, replacing it in the quadratic equation (it's given to you in the exercise)\\
[tex]x=\frac{-b+-\sqrt{b^{2}-4*a*c } }{2*a}[/tex]
[tex]x=\frac{-64+-\sqrt{64^{2}-4*-16*80 } }{2*-16}[/tex]
[tex]x=\frac{-64+-\sqrt{64^{2}+5120} }{-32}=\frac{-64+-\sqrt{4096+5120} }{-32}\\x=\frac{-64+-\sqrt{9216} }{-32}=\frac{-64+-\ 96 }{-32}\\x=\frac{-64+-\ 96 }{-32}\\x1=\frac{-64+\ 96 }{-32}=\frac{32}{-32} =-1\\\\x1=\frac{-64-\ 96 }{-32}=\frac{-160}{-32} =5\\[/tex]
We are working with a quadratic equation, so we will have two possible results,
[tex]X1=-1\\X2=5\\[/tex]
Now, we are asked to find the time, time can not be a negative value, we have to discard X1 (-1) since it's a negative value, knowing that, the rocket will hit the lake after 5 seconds.
Have a nice day!
