In a bacteria growing experiment, a biologist observes that the number of bacteria in a certain culture triples every 4 hours. After 20 hours, it is estimated that there are 2 million bacteria in the culture. How many bacteria were present initially?
The correct answer and explanation is:
To solve this, we can use the exponential growth formula for population dynamics: N(t)=N0×rt/TN(t) = N_0 \times r^{t/T}
Where:
- N(t)N(t) is the number of bacteria at time tt,
- N0N_0 is the initial number of bacteria,
- rr is the growth rate factor (in this case, the number of times the population triples),
- tt is the time elapsed,
- TT is the time it takes for one full growth cycle (in this case, 4 hours).
From the problem:
- N(t)=2,000,000N(t) = 2,000,000,
- t=20t = 20 hours,
- T=4T = 4 hours (because the population triples every 4 hours),
- r=3r = 3 (the population triples).
Now we can plug these values into the formula: 2,000,000=N0×320/42,000,000 = N_0 \times 3^{20/4}
Simplify the exponent: 2,000,000=N0×352,000,000 = N_0 \times 3^5
Calculate 353^5: 35=2433^5 = 243
So: 2,000,000=N0×2432,000,000 = N_0 \times 243
Solve for N0N_0 by dividing both sides by 243: N0=2,000,000243≈8231.68N_0 = \frac{2,000,000}{243} \approx 8231.68
Thus, the initial number of bacteria was approximately 8,232.
This exponential growth is typical for many biological systems, such as bacteria, where a fixed growth rate results in rapid increases in population size over time. The process assumes ideal conditions, with no environmental factors that would slow down growth, such as limited nutrients or overcrowding.