This problem comes from Illustrative Mathematics and can be used to explore exponential growth with students in a science context.

Here’s the problem:

On June 1, a fast growing species of algae is accidentally introduced into a lake in a city park. It starts to grow and cover the surface of the lake in such a way that the area covered by the algae doubles every day. If it continues to grow unabated, the lake will be totally covered and the fish in the lake will suffocate. At the rate it is growing, this will happen on June 30.

1. When will the lake be covered half-way?
2. On June 26, a pedestrian who walks by the lake every day warns that the lake will be completely covered soon. Her friend just laughs. Why might her friend be skeptical of the warning?
3. On June 29, a clean-up crew arrives at the lake and removes almost all of the algae. When they are done, only 1% of the surface is covered with algae. How well does this solve the problem of the algae in the lake?

Here’s a way to think about the answers to each question:

1. Since the algae cover doubles every day, if the lake is completely covered on June 30, then one day earlier, on June 29, half of the lake was covered.
2. On June 29, half the lake was covered. On June 28 one-fourth of the lake was covered. On June 27 one-eighth of the lake was covered. On June 26 only one sixteenth of the lake was covered by algae.When the friend does not believe the warning, only 1/16, or 6.25% of the water surface was covered, so 15/16=93.75% of the water surface was still open. The algae cover did not seem very dramatic at this point.
3. After the clean-up crew removes most of the algae, only 1% of the water surface is covered by algae. However, because these algae still grow just as quickly as before, the area covered still doubles every day. Therefore, after one day, 2% will be covered, after two days 4% will be covered, and so on. A table of values shows the following:

   number of days since clean-up	percentage of lake covered with algae
   0	1
   1	2
   2	2 2 = 4
   3	2 3 = 8
   4	2 4 = 16
   5	2 5 = 32
   6	2 6 = 64
   7	2 7 = 128

   Of course, it is impossible to cover more than 100% of the water surface, so the last entry in the table does not make sense. But we can conclude that sometime between day 6 and 7 the entire lake will be covered by algae. The lake clean-up will be undone completely in less than a week