A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
Question
A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
Solution
Sure, I can help with that. The problem describes exponential growth, which can be modeled by the equation P(t) = P0 * e^(kt), where:
- P(t) is the population at time t,
- P0 is the initial population,
- k is the growth rate,
- e is the base of natural logarithms (approximately 2.71828),
- and t is time.
Given that the initial population P0 is 100 cells and the population after one hour (t=1) is 420 cells, we can substitute these values into the equation to solve for k:
420 = 100 * e^(k*1)
First, divide both sides by 100:
4.2 = e^k
Then, take the natural logarithm of both sides to solve for k:
ln(4.2) = k
So, the growth rate k is approximately 1.43508.
This means that the size of the bacteria population increases by about 143.508% every hour.
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