Question

The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of 8.5% per hour. How many hours does it take for the size of the sample to double?

Answer 1

The formula for continuous exponential growth is:

N(t) = N₀e^(rt)

where:
N(t) is the size of the population at time t
N₀ is the initial size of the population
r is the growth rate
t is time

To find the time it takes for the population to double, we need to solve for t in the equation:

2N₀ = N₀e^(rt)

Dividing both sides by N₀, we get:

2 = e^(rt)

Taking the natural logarithm of both sides, we get:

ln(2) = rt

Solving for t, we get:

t = ln(2)/r

The growth rate is given as 8.5% per hour, which is equivalent to 0.085 per hour. Substituting this into the formula, we get:

t = ln(2)/0.085

t ≈ 8.14

Therefore, it takes approximately 8.14 hours for the size of the population to double.