IMPORTANT: If you want the "Attentiveness Rewarded" achievement/trophy, do not follow this route. Instead, use the shortcut shown here.

If you don't care about achievements, or just enjoy climbing around and pulling switches, try this longer route: Take a standing jump up onto the raised walkway running along the middle of the room. Head for the right end of the walkway and pull the switch to open the door at the other end.

Go through.

Just inside this room are 2 switches that raise a pair of hinged ledges on the opposite side of the room. They are timed, but it's a fairly slow timer, about 30 seconds for each ledge, so there's no need to panic. To get across, pull the switch near the door...

Press Look to cancel the cutscene.

Side flip to the left, pull the second switch...

...and press Look again.

Then roll and take a running jump across the room...

...to grab the first hinged ledge.

Traverse to the right above a pit lined with broken glass. Don't hesitate until you've passed the hinged ledges and don't drop down on the other side. Continue traversing to the right and pull up into the corner alcove.

Here you'll find another switch.

Use it to lower another hinged ledge back in the room with the ducts. This one is not timed.

Drop to the floor, cross the room to the low ramp, climb onto the ledge with the switches, and re-enter the big room where you started.

Cross to the far left corner. Now that you've lowered the hinged ledge here, you can use the angled metal blocks to climb higher.

Stand facing the lower angled block in the corner. Step back until Lara bumps into the other angled block behind her.

Then jump forward onto the first slope.

Continue holding Jump and Action, and Lara will backflip onto the higher block, spring forward, and grab the corner ledge above.

Pull up, turn around, and take a running jump over the gap where the hinged ledge was.

Follow the duct to the top, where you'll find a movable block.

Pull the block to the right...

...so you can then move around behind it.

Hang from the duct where the box was, traverse left, and pull up.

Turn around, walk to the edge, and take a standing jump to grab the ledge ahead.

Pull up and follow the dark passageway to the end. Slide down the ramp on the right...

...and proceed along the top of the ducts...

...until you can jump into the doorway with the blue wood paneling. Pick up the main walkthrough with the section "PASSENGER AREAS."
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