Analysis:
To find the maximum height reached by the rocket, we need to consider the rocket's motion in two phases: the powered flight phase and the unpowered flight phase.

Phase 1: Powered Flight
In the powered flight phase, the rocket is accelerating vertically upwards due to the thrust of the engines. The given information states that the resultant vertical acceleration during this phase is 10 m/s^2.

Using the equation of motion:
v = u + at

where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken, we can find the time taken for the fuel to be finished.
Since the rocket is fired vertically upwards, the initial velocity (u) is 0 m/s. The final velocity (v) can be calculated as follows:

v = u + at
v = 0 + (10 m/s^2)(60 s) (since 1 minute is equal to 60 seconds)
v = 600 m/s

Therefore, the final velocity when the fuel is finished is 600 m/s.

Phase 2: Unpowered Flight
Once the fuel is finished, the rocket enters the unpowered flight phase. In this phase, the only force acting on the rocket is gravity, which causes it to decelerate. The rocket will continue to move upwards until its velocity becomes zero, at which point it will start to fall back down.

Using the equation of motion:
v^2 = u^2 + 2as

where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement, we can find the maximum height reached by the rocket.

Since the final velocity (v) is zero when the rocket reaches its maximum height, the equation becomes:
0 = u^2 + 2as

We know the initial velocity (u) is 600 m/s and the acceleration (a) is -10 m/s^2 (negative because it is acting in the opposite direction to the initial motion).

0 = (600 m/s)^2 + 2(-10 m/s^2)s

Simplifying the equation, we can solve for the displacement (s):

0 = 360,000 m^2/s^2 - 20s
20s = 360,000 m^2/s^2
s = 18,000 m

Therefore, the maximum height reached by the rocket is 18,000 meters.

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