Mathematics · Middle school

Exponential growth: the lily pond

If the leaves double every day, on which day is the pond half full? Compare linear, exponential and logistic growth side by side.

In the learning section below the figure, predict, try, complete the tasks and check yourself with short questions.

Fig. 1The lily pond

Move the day forward or press play: three ponds start the same and grow by different rules

day 7 · Exponential: 12.5%

Day tday 7

Daily factor RR = 2

Your own values

R (factor)

Daily growth

Start N₀

Length (days)

Day t

R from 1.001 to 10 (or type the daily growth as a percentage), at most 120 days, at most 100 leaves to start. A value applies when you press Enter or leave the box.

Today: day 7

T₂ = 1 days

Linearsame amount each day	8	0.781%
Exponentialsame proportion each day	128	12.5%
Logisticslows as the pond fills	114	11.1%

Daily growth: 100%
Day the exponential pond fills: 10
Day it is half full: 9
Day the logistic pond is half full: 10

A pond holds 1,024 leaves (2¹⁰ = 32 × 32).

Leaves – days

Learn

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1Predict
2Try
3Explain

The leaves double every day and the pond is completely covered on day 10. On which day was it half covered?

In exponential growth each day adds the same proportion, not the same amount: N = N₀ × Rᵗ. That is why for a long time it looks as if nothing is happening, and then a few doublings change everything; one doubling time before the pond is full (one day when R = 2) it is still half empty. Nothing can grow exponentially for ever: when space, food or resources run out, growth slows. The logistic curve is the simplest model of that. This is an abstract model, not a forecast for a real pond, disease or population. See A. A. Bartlett, “Forgotten Fundamentals of the Energy Crisis”, Am. J. Phys. 46, 876 (1978; his lecture is titled “Arithmetic, Population and Energy”); Verhulst’s logistic equation (1838).

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