15. If a bacterial culture with a doubling time of 30 minutes starts with two cells, then the number of cells after 4 hours are ______.
Bacterial Growth Calculation: Number of Cells After 4 Hours
Introduction
Bacterial populations increase through binary fission, a process in which one bacterial cell divides into two genetically identical daughter cells. Under ideal environmental conditions, every daughter cell continues dividing at regular intervals, producing an exponential increase in the total population. Because of this predictable pattern, bacterial growth can be described mathematically using exponential equations.
Correct Answer
Correct Answer: 512 Cells
Detailed Explanation
The bacterial population doubles every 30 minutes. Therefore, the first step is to calculate the total number of generations that occur in 4 hours.
Step 1: Convert hours into minutes
4 hours = 4 × 60 = 240 minutes
Step 2: Calculate the number of generations
Number of generations = Total Time ÷ Doubling Time
n = 240 ÷ 30 = 8 generations
Step 3: Apply the bacterial growth formula
N = N0 × 2n
Where:
- N = Final number of cells
- N0 = Initial number of cells
- n = Number of generations
Given:
N0 = 2 cells
n = 8
Therefore,
N = 2 × 28
N = 2 × 256
N = 512 cells
Thus, after 4 hours, the bacterial culture will contain 512 cells.
Step-by-Step Calculation
| Parameter | Value |
|---|---|
| Initial Cells | 2 |
| Doubling Time | 30 minutes |
| Total Time | 240 minutes (4 hours) |
| Number of Generations | 240 ÷ 30 = 8 |
| Growth Formula | N = N0 × 2n |
| Final Calculation | 2 × 28 = 512 |
Generation-wise Growth of the Culture
| Generation | Time (Minutes) | Number of Cells |
|---|---|---|
| 0 | 0 | 2 |
| 1 | 30 | 4 |
| 2 | 60 | 8 |
| 3 | 90 | 16 |
| 4 | 120 | 32 |
| 5 | 150 | 64 |
| 6 | 180 | 128 |
| 7 | 210 | 256 |
| 8 | 240 | 512 |
Important Formula Used in Bacterial Growth
| Formula | Application |
|---|---|
| N = N0 × 2n | Final bacterial population |
| n = Time ÷ Generation Time | Number of generations |
| Generation Time = Total Time ÷ Number of Generations | Doubling time calculation |
| Growth Rate (k) = n ÷ Time | Determining growth rate constant |
Why Bacterial Growth is Exponential
During binary fission, every bacterial cell divides into two daughter cells. Each daughter cell then undergoes another division after one generation time. Consequently, the number of cells doubles with every generation, producing an exponential growth pattern represented by powers of two. This mathematical relationship explains why bacterial populations can increase from only a few cells to millions within a relatively short period under optimal growth conditions.
Applications of Bacterial Growth Calculations
| Field | Application |
|---|---|
| Industrial Biotechnology | Optimization of fermentation processes |
| Clinical Microbiology | Prediction of bacterial multiplication during infection |
| Food Microbiology | Estimating food spoilage and shelf life |
| Environmental Microbiology | Monitoring microbial population dynamics |
| Molecular Biology | Designing bacterial culture experiments |
Final Answer
Initial Cells = 2
Doubling Time = 30 minutes
Total Time = 4 hours = 240 minutes
Number of Generations = 8
Final Number of Cells = 2 × 28 = 512
Correct Answer: 512 Cells
