Final answer:
The phase of the cell cycle associated with rapid protein synthesis and growth to double the cell's size is the G1 phase, which is typically the longest phase of the cycle and focused on cell growth and production of proteins necessary for DNA replication and cell division.
Step-by-step explanation:
The phase of the cell cycle during which rapid protein synthesis occurs as the cell grows to double its size is the G1 phase (option B). During the G1 phase, also known as Gap 1, the cell undergoes rapid growth and prepares for DNA replication by synthesizing enzymes and other proteins. This phase focuses on cell growth and the production of the molecular machinery necessary for the subsequent phases of the cell cycle. After completing the G1 phase, if the cell is set to divide, it will enter the S phase where DNA replication occurs, followed by the G2 phase where the cell continues to grow and makes final preparations for mitosis. The G1 phase is where the most significant increase in size and protein synthesis takes place as the cell readies itself for division.