Answer:
4 minutes 31.06 seconds
Explanation:
We can start by converting the speed and distance to units compatible with the acceleration.
12.42 mi = (12.42 mi)(5280 ft/mi) = 65,577.6 ft
203 mi/h = (203 mi/h)(22/15 ft/s)/(1 mi/h) = 297 11/15 ft/s = 4466/15 ft/s
Then the time taken to reach top speed will be ...
t = v_f/a = (4466/15 ft/s)/(2.93 ft/s^2) = 101.615 s
The distance covered while reaching top speed is ...
d = v^2/(2a) = (4466/15 ft/s)^2/(2(2.93 ft/s^2)) = 15,127.157 ft
So, the remaining distance is ...
d_remaining = 65,577.6 ft -15,127.157 ft = 50,450.443 ft
And the time required to cover that distance is ...
t = d_remaining/v_f = (50,450.443 ft)/(4466/15 ft/s) = 169.448 s
The total time taken for acceleration and cruise to the top is ...
101.615 s + 169.448 s = 271.064 s ≈ 4 minutes 31.06 seconds
_____
We have assumed that the rocket's acceleration is fully applied in the direction of travel. We have to assume that the acceleration due to gravity is otherwise accounted for.