Question

the bottle is completely full at 11:40 a.m. which statements help explain this result? check all that apply. the bottle was 1/2 full at 11:35 a.m. and doubled again after 5 minutes. the bottle was 1/4 full at 11:30 a.m. and doubled twice after 10 minutes. the bottle was 1/2 full at 11:35 a.m. and more bacteria were added to fill the bottle. exponential growth involves a constant multiplicative rate of change. exponential growth involves a constant additive rate of change.

Explanation:

Step1: Analyze time - volume relationship

If the bottle is full at 11:40 a.m., and it was 1/2 full at 11:35 a.m., in 5 minutes (from 11:35 a.m. to 11:40 a.m.) the amount of liquid doubled since $\frac{1}{2}\times2 = 1$ (full - bottle). So the first statement is correct.

Step2: Analyze another time - volume relationship

If the bottle was 1/4 full at 11:30 a.m., after 10 minutes (by 11:40 a.m.), it doubles twice. Starting from $\frac{1}{4}$, after the first double it becomes $\frac{1}{4}\times2=\frac{1}{2}$, and after the second double it becomes $\frac{1}{2}\times2 = 1$ (full - bottle). So the second statement is correct.

Step3: Recall the nature of exponential growth

Exponential growth is characterized by a constant multiplicative rate of change, not an additive rate of change. For example, if the population of bacteria is growing exponentially, it multiplies by a certain factor over a fixed time - interval, not adding a fixed amount. So the fourth statement is correct and the fifth statement is incorrect. The third statement is incorrect because it implies external addition instead of natural growth.

Answer:

The bottle was 1/2 full at 11:35 a.m. and doubled again after 5 minutes.
The bottle was 1/4 full at 11:30 a.m. and doubled twice after 10 minutes.
Exponential growth involves a constant multiplicative rate of change.