# When Was the Petri Dish Half Full?

Imagine a scientist places a single bacterium in a petri dish at exactly noon. This bacterium has the ability to divide into two every minute. By 1:00 PM, the petri dish is completely full of bacteria. The question that arises is, at what time was the petri dish half full? This seemingly simple question can be quite perplexing, but it provides a fascinating insight into the concept of exponential growth. Let’s delve into this intriguing scenario and understand the mathematics behind it.

## Understanding Exponential Growth

Exponential growth refers to the process where the growth rate of a mathematical function is proportional to the function’s current value. In our scenario, the bacteria’s population doubles every minute, which is a classic example of exponential growth. This means that the number of bacteria at any given minute is twice the number of bacteria in the previous minute.

## The Half Full Petri Dish

Given that the petri dish is full at 1:00 PM, and knowing that the bacteria’s population doubles every minute, it’s logical to conclude that the petri dish was half full just one minute before it became completely full. Therefore, the petri dish was half full at 12:59 PM. This is because, in the next minute, the number of bacteria would double, filling the petri dish completely.

## Implications of Exponential Growth

While this scenario is a simple mathematical problem, it has profound implications in various fields such as biology, finance, and computer science. In biology, understanding exponential growth can help scientists predict the spread of diseases or the growth of populations. In finance, it’s used to calculate compound interest. In computer science, it’s used in algorithms and data structures.

## Common Misconceptions

One common misconception is that the petri dish would be half full at 12:30 PM, halfway through the hour. However, this is incorrect due to the nature of exponential growth. The growth is not linear, but rather doubles with each passing minute. Therefore, the petri dish is not half full halfway through the hour, but just a minute before the dish becomes completely full.

## Conclusion

The question of when the petri dish was half full provides a fascinating look into the concept of exponential growth. It shows how quickly things can multiply under the right conditions, and how our intuition can sometimes lead us astray when dealing with exponential processes. So, the next time you encounter a situation involving exponential growth, remember the bacteria in the petri dish and think twice before making assumptions.