Question

the bottle is completely full at 11:40 a.m.
which statements help explain this result? choose three correct answers.
the bottle was \\(\frac{1}{4}\\) full at 11:30 a.m. and doubled twice after 10 minutes.
exponential growth involves a constant multiplicative rate of change.
the bottle was \\(\frac{1}{2}\\) full at 11:35 a.m. and more bacteria were added to fill the bottle.

Explanation:

1. For the first statement: From 11:30 to 11:40 (10 minutes), if the bottle was $\frac{1}{4}$ full at 11:30, doubling twice gives $\frac{1}{4} \times 2 \times 2 = 1$, meaning full at 11:40, which matches the given result.
2. For the second statement: Bacterial growth is exponential, defined by a constant multiplicative (percent) growth rate, which is the core reason for this doubling pattern.
3. For the third statement: At 11:35 (5 minutes before full), the bottle was $\frac{1}{2}$ full. The last $\frac{1}{2}$ of the bottle is filled in the final 5 minutes, which is more bacteria than were present in the first 35 minutes combined, aligning with exponential growth behavior.

Answer:

• The bottle was $\frac{1}{4}$ full at 11:30 a.m. and doubled twice after 10 minutes.
• Exponential growth involves a constant multiplicative rate of change.
• The bottle was $\frac{1}{2}$ full at 11:35 a.m. and more bacteria were added to fill the bottle.