Question

Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 1900 bacteria selected from this population reached the size of 1982 bacteria in one and a half hours. Find the hourly growth rate parameter

Answer 1

In continuous exponential growth, the size of a population can be modeled by the formula:

P = P0 * e^(rt)

where:

P = final amount (1982 bacteria in this case)

P0 = initial amount (1900 bacteria in this case)

r = growth rate (the variable we're trying to find)

t = time (1.5 hours in this case)

First, let's isolate the variable r in the equation:

1982 = 1900 * e^(1.5r)

Dividing both sides by 1900 gives:

1982 / 1900 = e^(1.5r)

Taking the natural logarithm (ln) of both sides (since the natural logarithm is the inverse of the exponential function) gives:

ln(1982 / 1900) = 1.5r

Now, solve for r:

r = ln(1982 / 1900) / 1.5

Using a calculator to evaluate this, we get:

r ≈ 0.0224

Therefore, the hourly growth rate is approximately 0.0224 or about 2.24% per hour. This means the bacteria population is growing at a rate of about 2.24% every hour.