Final answer:
The correct path of protein production and secretion in a eukaryotic cell is from the rough endoplasmic reticulum to the Golgi apparatus and then to the cell membrane.
Step-by-step explanation:
The correct answer is A. Rough endoplasmic reticulum → Golgi apparatus → cell membrane.
Protein production and secretion in a eukaryotic cell begins in the rough endoplasmic reticulum (RER), where ribosomes on the surface of the RER synthesize proteins. The newly synthesized proteins remain bound to the membrane of the RER and are then transported in transport vesicles to the Golgi apparatus. The Golgi apparatus modifies and packages the proteins into secretory vesicles, which then fuse with the cell membrane and release the proteins to the extracellular environment. Here, proteins undergo further modifications and sorting. Finally, they are packaged into secretory vesicles that move towards and fuse with the plasma membrane, leading to secretion out of the cell or incorporation into the cell membrane.