If a bacterial culture initially contains 4 bacteria and has a generation time of 30 minutes, how many bacteria will be present in the culture in 90 minutes? Assume that all bacteria survive.
The correct answer and explanation is:
Let’s break down the problem step-by-step:
Problem:
- Initial number of bacteria, N0=4N_0 = 4
- Generation time = 30 minutes (time it takes for the bacteria population to double)
- Total time = 90 minutes
- Assume all bacteria survive and multiply
What is being asked?
Find the total number of bacteria NN present after 90 minutes.
Step 1: Calculate the number of generations
Since bacteria double every generation time, the number of generations nn in 90 minutes is: n=total timegeneration time=9030=3 generationsn = \frac{\text{total time}}{\text{generation time}} = \frac{90}{30} = 3 \text{ generations}
Step 2: Use the formula for bacterial growth
The bacterial population after nn generations is given by: N=N0×2nN = N_0 \times 2^n
Substitute the values: N=4×23=4×8=32N = 4 \times 2^3 = 4 \times 8 = 32
Answer:
After 90 minutes, the bacterial culture will contain 32 bacteria.
Explanation (300 words):
Bacteria reproduce through a process called binary fission, where one bacterial cell divides into two identical daughter cells. The time it takes for the bacterial population to double is known as the generation time or doubling time. In this case, the generation time is 30 minutes, meaning every 30 minutes the population doubles in size.
Starting with 4 bacteria, after the first 30 minutes (one generation), the number of bacteria doubles from 4 to 8. After another 30 minutes (60 minutes total, or two generations), the population doubles again from 8 to 16. After a third 30-minute period (90 minutes total), it doubles once more from 16 to 32.
Mathematically, the growth can be modeled using the exponential growth formula: N=N0×2nN = N_0 \times 2^n
Where:
- N0N_0 is the initial number of bacteria,
- nn is the number of generations (total time divided by generation time),
- NN is the number of bacteria after nn generations.
Because bacterial growth is exponential under ideal conditions (no nutrient limitations, no death), this model assumes all bacteria survive and divide at a constant rate.
Hence, after 90 minutes, which encompasses 3 full generations, the bacterial population grows from 4 to 32 cells. This exponential increase illustrates why bacterial cultures can rapidly multiply, highlighting the importance of generation time in microbiology and medicine.