How Long Does It Take for a Lily Pad to Cover a Quarter of a Pond?

Imagine a lily pad in a pond, growing at a remarkable rate. Every minute, its size doubles, leading to an interesting question: if it takes one hour (60 minutes) for the lily pad to cover the entire pond, how long does it take to cover just a quarter of the pond?

Understanding the Growth Pattern

The growth pattern of the lily pad is exponential. Every minute, its size doubles, creating a geometric progression (G.P.). Let's denote the initial area of the lily pad as x. Here’s how it progresses over each minute:

After 1 minute: 2x After 2 minutes: 4x After 3 minutes: 8x And so forth, up to 60 minutes where the area is 259x, covering the entire pond.

Backward Calculation

To determine when the lily pad covers a quarter of the pond, let's work backward from the moment it covers the entire pond:

At 60 minutes, the pond is fully covered (area is 259x). At 59 minutes, the lily pad covers half of the pond (area is 258x). At 58 minutes, the lily pad covers a quarter of the pond (area is 257x).

So, the lily pad takes 58 minutes to cover a quarter of the pond.

Alternative Explanation

Another way to look at it is to recognize that the lily pad doubles in size every minute. Therefore, it only takes 2 minutes to go from a quarter of the pond to the entire pond. Working backward:

60 minutes: Full pond 59 minutes: Half pond 58 minutes: Quarter pond

This confirms that it takes 58 minutes for the lily pad to cover a quarter of the pond.

Mathematical Representation

Mathematically, we can express the area of the lily pad at different times as a geometric progression:

Initial area: x After 1 minute: 2x After 2 minutes: 4x 22x After 3 minutes: 8x 23x And so on, up to 60 minutes: 26 (full pond)

To find the quarter of the pond, we need the term in the geometric progression where the area is 257x (one-quarter of the full pond), which corresponds to 58 minutes.

Conclusion

In conclusion, it takes 58 minutes for the lily pad to cover a quarter of the pond. This problem demonstrates the power of exponential growth and the importance of understanding geometric progressions.