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.

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.