Question

During the exponential growth, it took 6 hours for the population of bacterial cells to increase from \(2.5 \times 10^6\) to \(5 \times 10^{8}\). The generation time of the bacterium, rounded off to the nearest integer, is _______ minutes.

Hint:

Always compute the number of generations first using $\log(N_t/N_0)/\log 2$, then divide total time by this to get the generation time.

Answer 1

Step 1: Formula. Generation time $g = \dfrac{t}{n}$, where $n$ = number of generations.
Step 2: Calculate $n$. \[ n = \frac{\log(N_t) - \log(N_0)}{\log 2} \] \[ = \frac{\log(5 \times 10^8) - \log(2.5 \times 10^6)}{\log 2}. \] \[ = \frac{\log\left(\frac{5 \times 10^8}{2.5 \times 10^6}\right)}{\log 2} = \frac{\log(200)}{\log 2}. \] \[ \log(200) \approx 2.3010, \log 2 \approx 0.3010. \] \[ n \approx \frac{2.3010}{0.3010} \approx 7.64. \] Step 3: Calculate generation time. Total time = 6 h = 360 min.
\[ g = \frac{360}{7.64} \approx 47.1 \text{ min}. \] Rounded to nearest integer = 47 min.