The Half-Covered Mystery of the Water Lily

Have you ever pondered a seemingly simple question that involves a fascinating mathematical concept? One such question involves a water lily in a pond that doubles its size every day. Here, we will explore the mystery of when the pond is half-covered, unravel the intricacies of geometric growth, and highlight the power of exponential increase.

Understanding the Growth Pattern

The question at hand is, ‘A water lily in a pond doubles its size every day. In 28 days, the flower will cover the entire pond. Question: In how many days will the pond be half-covered? ’ This question is a prime example of geometric growth, a fundamental concept in both mathematics and real-world scenarios such as population growth and technological advancements.

Why the Pond is Half-Covered on Day 27

If the patch of lilies doubles every day, the growth pattern follows a sequence of 2, 4, 8, 16, and so on. Logically, if the pond is fully covered in 28 days, then it would have covered half of the pond the day before, on day 27. This is because the lily patch doubles in size each subsequent day, and therefore, by working backward, we can deduce that the pond was half-covered a day before its complete coverage.

Mathematical Explanation of the Problem

The growth of the lily patch can be described mathematically. If we denote the number of lilies as N, the pond is fully covered on day 28, which means that N on day 28 is the maximum capacity of the pond. Since the patch doubles every day, on day 27, the number of lilies would be half of the pond's capacity, i.e., N/2. This can be expressed as:

N on day 28 2 × (number of lilies on day 27)

This implies that the number of lilies on day 27 is N/2, indicating that the pond was half-covered on day 27. This is a direct application of the properties of geometric growth, where the population (or in this case, the lily patch) doubles every time period.

Geometric Progression and Its Application

The growth of the lily patch follows a geometric progression with a common ratio of 2. If we start with 1 lily on the first day, the sequence of the number of lilies would be 1, 2, 4, 8, and so on. By the 10th day, the pond is half-covered. Since the lily patch doubles every day, it would take just one more day to cover the entire pond. Therefore, if the pond is fully covered on the 20th day, it would be half-covered on the 19th day.

Conclusion

The answer to the question 'In how many days will the pond be half-covered?' is 27 days. This observation underscores the power of exponential growth, where small beginnings can lead to dramatic outcomes over time. Whether it’s the growth of a lily patch in a pond, the spread of an idea, or the growth of a technology, understanding and predicting this type of growth is crucial in many fields.