The square doesn't need to be in the middle; it just needs to be in a position such that the entire square is in the magnetic field. This is because the magnetic field is uniform.
Remember, the formula flux = BA refers to the area that is exposed to the magnetic field.
Now, the maximum voltage induced would then be due to the change in flux from when the square first enters the field to when it is fully immersed in the field. The side length is 2 cm, and the speed of motion is 4 cm/s, so the time we want is 0.5 s.
Just one loop only.
The change in flux is 3.7*10^-3 T * (2 cm)^2 = 14.8 * 10^-7 T m^2 (note change in units)
So dividing by the time, our max voltage is 29.6* 10^-7 V, so 2.96 uV which is the unrounded form of the answer given.
The trick is identifying what the time means. The time is the period of time over which the change in flux occurs. Note that when the square is in the middle of the magnetic field, the entire square will have remained in the field for a period of time, so the flux isn't changing; the voltage is zero in that case.
Honestly mate, I don't fully understand half the crap we're learning in physics and I got 96% on that sample VCAA 2013 exam.
All you need to do is sub in values.
We know speed = distance/time so therefore time = distance/speed = 0.02/0.04 = 0.5, all values directly from the question subbed into our formulas from our A3 cheat sheet.
Then sub time = 0.5 into the EMF equation and wallah, you get the answer.
Another example of how physics is becoming ridiculous; you don't even need any understanding of the mechanics of the formulas to get the marks.