Amodel for the path of a toy rocket is given by h= -68t - 4.9t^2 how long does it take the rocket to hit the ground? how high was the rocket at 2 seconds? what is the maximum height of that rocket and how long does it take that rocket to reach it's maximum height?

The right equation has to be h= 68t - 4.9t^2, or you will get negative times.

I will show you the solution using h = 68t - 4.9t^2

1) How long does it take the rocket to hit the ground?

ground level => h = 0

=>  68t - 4.9t^2 = 0

factor the polynomial => - t (4.9t - 68) = 0

=> t = 0 (which is when the rocket takes off) and 68 + 4.9t = 0 (which is when the rocket to return and hit the ground).

4.9t - 68 = 0 => t = 68 / 4.9 = 13.88 s

13.88 s

2) How high was the rocket at 2 seconds?

replace t = 2 in the function h = - 4.9t^2 + 68

=> h = -4.9 (2^2) + 68 = 48.4 m

48.4 m

3) What is the maximum height of that rocket and how long does it take that rocket to reach it's maximum height?

Find the vertex of the parabola.

You should know that the vertex is at the midpoint between the two roots (zeros of the polynomial).

You already found the zeros, t = 0 and t = 13.88

So0 the vertex is at t = [0 + 13.88] / 2 = 6.94 s

And the maximum height is h = - 4.9 (6.94)^2 + 68 = 168 m

h max = 168 m, t = 6.94 s.