Would the trachea collapse when you were exhaling if there were no rings of cartilage in its walls

Yes, the trachea would collapse in this situation.
The trachea is a tube composed of cartilage which connects the pharynx and larynx to the lungs. It is a critical structure for all air-breathing animals and its main responsibility is to facilitate the passage of air. The trachea is surrounded by several c-shaped and incomplete cartilaginous rings which prevent it from collapsing.

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