What component controls many functions of the cell, including protein synthesis?

The nucleus is the correct answer because it serves as the control center of the cell, containing the cell's genetic material, DNA. This genetic material holds the instructions for various cellular processes, including protein synthesis. In eukaryotic cells, the process of protein synthesis begins in the nucleus, where mRNA is transcribed from DNA. The mRNA then leaves the nucleus and is translated into proteins by ribosomes in the cytoplasm.

While the rough endoplasmic reticulum is also involved in protein synthesis, it does not control the overall function of the cell as the nucleus does. The rough endoplasmic reticulum is specifically responsible for the modification and transport of newly synthesized proteins, but it relies on instructions from the nucleus. The vacuole is primarily responsible for storage and maintaining cell turgor pressure, and the cell wall provides structural support but does not play a direct role in controlling cellular functions. Thus, the nucleus is essential for regulating the cell's activities, including the synthesis of proteins.