The cylinder moves downwards a distance x. How far up does the box move in that time?

It should be a simple string length question. No forces are needed.
The string length is constant, if the left part and middle parts are longer (how much?), how much shorter is the right part?

edit - they don't have the same acceleration.

It should be a simple string length question. No forces are needed.
The string length is constant, if the left part and middle parts are longer (how much?), how much shorter is the right part?

edit - they don't have the same acceleration.

I'm not the OP and not looking for a solution, but this looks interesting as it's a bit trickier than the usual "two objects hanging over a pulley" questions!

I can see where the 2x distance has come from, but can I just check - is the OP's diagram correct apart from the assumption that the accelerations are equal?

So our assumption in solving this is that the tape's tension is constant throughout, and without further information we can only write the box's acceleration and the overall tension in terms of a. Am I thinking this out correctly?

Part c) says the tape tension is constant throughout?

One of these days we might even get a block and tackle question on here - unless they're all in physics help.

As memory serves, "smooth pulley" is the magic phrase that says "you can assume the tension is constant along the string".

I thought Step I, 2014, Q11 was quite a fun pulley question.

The left object goes down by x. The left and middle substring must be x longer each. To the OP, you need differences, not proportions. They rob this from the right substring which must be 2x shorter.

Parts b&c bring tensions and accelerations into the probkem. It's not needed for a). Part c) says the tape tension is constant throughout?

Obviously this is the basis for pulleys, where the work is
Force*distance
So you use a pulley to apply a higher force over a shorter distance for the same work done. See pulley - mechanical advantage examples in
https://www.bbc.co.uk/bitesize/guides/zbt26yc/revision/6

This info on pulleys wasn't covered at all in my textbook. Is it common knowledge or are they assuming I study physics?

That link was to gcse physics.
Tbh, it's the simplest such pulley and it's not unreasonable that you could apply the stuff you learnt on a level mechanics to a simple gcse-type problem. The question tries to walk you through the approach by getting you to think about distances, then accelerations, then tension. Rather than the other way round. So a bit unusual but not unreasonable.

I don't remember covering pulleys in that much depth in my GCSEs. I looked up the word in my old Edexcel CGP physics textbook and it only appears 5 or 6 times in total. The link you provided is for the AQA GCSE spec, so maybe that is why? Looks like I need to get the AQA CGP textbook now...lol