1) For a rotating object, we can write the analogous of Newton's second law as

where

is the torque applied to the object, I is the moment of inertia and

is the angular acceleration.
2) We can write the ratio between the starting torque and the stopping torque as:

3) Then, we can rewrite the two angular accelerations. The general formula is

where

and

are the final and initial angular velocities, while t is the time.
4) Let's start by calculating the starting acceleration. In this case, the initial velocity is zero, while the final velocity is the full rotational speed (let's call it

, and the time is t=5.0 s:

When the carousel stops, instead, the initial velocity is

while the final velocity is 0, and the time is t=11.0 s:

Therefore, if we replace the two angular accelerations that we found into the formula of the torque ration written at step 2), we find:

where the negative sign simply means that the two torques have opposite direction. Therefore, the ratio between starting and stopping torque is 11:5.