The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of $5 \%$ per hour. How many hours does it take for the size of the sample to double?
Note: This is a continuous exponential growth model.
Do not round any intermediate computations, and round your answer to the nearest hundredth.
Answer
Rounding to the nearest hundredth, we find that it takes approximately \(\boxed{13.86}\) hours for the size of the sample to double.
Step 1 ：We are given a continuous exponential growth model, which is represented by the formula \(N(t) = N_0 * e^{rt}\), where \(N(t)\) is the final amount, \(N_0\) is the initial amount, \(r\) is the growth rate, and \(t\) is the time.
Step 2 ：In this problem, we want to find the time it takes for the population to double. This means that \(N(t) = 2 * N_0\).
Step 3 ：We can set up the equation \(2 = e^{0.05t}\) and solve for \(t\).
Step 4 ：By solving the equation, we find that \(t = 13.862943611198904\).
Step 5 ：Rounding to the nearest hundredth, we find that it takes approximately \(\boxed{13.86}\) hours for the size of the sample to double.
