A bacterial cell divides once every minute. It takes one hour to fill a cup. How much time will it take to fill half the cup ?

Question

Accepted Answer

The correct option is 2:

This problem is a classic illustration of **exponential growth**, which is a core concept in biology, particularly when studying bacterial populations. The key piece of information is the doubling time: the bacterial cell divides once every minute, which means the entire population, and consequently the volume it occupies in the cup, **doubles** every minute. The problem states that the cup is completely full at the end of exactly one hour, which is 60 minutes. Therefore, at 60 minutes, the cup holds 100% of its capacity. For the cup to be half full (50% capacity), we must look back only one doubling period. Since the volume doubles every minute, the state of being half full must occur exactly one minute *before* the state of being completely full. If the population at time $t$ is $N(t)$, and the cup is full when $N(60) = N\_{full}$, then $N(59)$ must be exactly half of $N(60)$ because $N(t+1) = 2 \cdot N(t)$. Subtracting one minute from the total fill time gives the time required to fill half the cup: $60 \text{ minutes} - 1 \text{ minute} = 59 \text{ minutes}$.

The incorrect option is 1:

The option of **29 Minutes** is based on an incomplete or misunderstood application of the doubling principle. A student might arrive at this number by assuming a linear or arithmetic progression, perhaps miscalculating the number of minutes or confusing the starting point of the measurement. In the context of bacterial growth, if the cup were half full at 29 minutes, it would be completely full at 30 minutes (since $29 \text{ minutes} + 1 \text{ minute} = 30 \text{ minutes}$ is one doubling period). However, the problem explicitly states that the cup is full at 60 minutes. Therefore, for the population to double from half-full at 29 minutes to full at 30 minutes would contradict the given information that the full time is one hour.

The incorrect option is 3:

Selecting **30 Minutes** is the most common mistake for this type of question, as it is based on the assumption of **linear growth** rather than exponential growth. Students mistakenly calculate half of the total time ($60 \text{ minutes} / 2 = 30 \text{ minutes}$) under the belief that the bacteria accumulate at a constant rate, like water filling a tub. This calculation fails to account for the fact that the rate of accumulation is constantly increasing because the number of bacteria itself is doubling every minute. Since the final doubling event is what takes the cup from half-full to full, half the time only yields a small fraction of the total required population size, not half the volume.

The incorrect option is 4:

The answer **40 Minutes** does not correspond to any logical or conceptual step in either exponential or linear growth models. If the cup were half full at 40 minutes, it would mean the population doubled from 40 minutes to 41 minutes. This would mean that in the remaining 20 minutes (from 40 to 60), the cup would have to double its volume 20 times (a factor of $2^{20}$), which is an astronomically large and incorrect result based on the problem's premise. This choice is an arbitrary guess that ignores the defining characteristic of the problem, which is the consistent, single-minute doubling time of the bacterial population.