Question

A population of bacterial cells grows from 10,000 to 100,000,000 cells in 6 hours. The generation time of the bacterial population is ........... min. (rounded off to 2 decimals)

Hint:

The generation time can be calculated using the formula \( N_t = N_0 \times 2^n \), where \( n \) is the number of generations. Divide the total time by the number of generations to get the generation time.

Answer 1

Step 1: Using the exponential growth formula.
The exponential growth formula for a population is given by: \[ N_t = N_0 \times 2^n \] Where:
- \( N_t \) is the final population size
- \( N_0 \) is the initial population size
- \( n \) is the number of generations
We are given:
- \( N_0 = 10,000 \)
- \( N_t = 100,000,000 \)
- Time = 6 hours
We need to find \( n \), the number of generations: \[ 100,000,000 = 10,000 \times 2^n \] \[ 2^n = \frac{100,000,000}{10,000} = 10,000 \] Taking the logarithm of both sides: \[ n \log 2 = \log 10,000 \] \[ n = \frac{\log 10,000}{\log 2} = \frac{4}{0.3010} \approx 13.29 \, \text{generations} \] Step 2: Calculating generation time.
The generation time \( T \) is the time per generation: \[ T = \frac{\text{Total time}}{n} = \frac{6 \, \text{hours} \times 60 \, \text{minutes/hour}}{13.29} \] \[ T = \frac{360}{13.29} \approx 27.1 \, \text{minutes} \] Final Answer: \[ \boxed{27.1 \, \text{min}} \]