Q.71 A rich medium is inoculated with a bacterium that divides every
30 minutes. The number of bacteria at the end of 50 hours is
Options:
| (A) 2 × 1010 | (B) 2 × 1020 | (C) 1 × 1050 | (D) 1 × 2100 |
Answer: 2 × 1020 (Option B)
The number of bacteria after 50 hours is 2 × 1020.
Question Breakdown
Bacterial growth follows exponential growth: N = N₀ × 2n where doubling time = 30 minutes.
Calculate generations: 50 hours = 50 × 60 / 30 = 100 generations.
Starting from 1 cell: N = 2100 ≈ 1.267 × 1030. Options represent different approximations.
Step-by-Step Calculation
Time = 50 hours = 3,000 minutes Generations n = 3,000 / 30 = 100 doublings N = 2100 = (210)10 = (1,024)10 ≈ (103)10 = 1030 Precise: 2100 = 1.26765 × 1030
Log conversion: log10(2100) = 100 × log10(2) = 100 × 0.3010 = 30.10 → 1030
Option Analysis
| Option | Generations | Calculation | Matches 100? |
|---|---|---|---|
| (A) | ~33 | 233 ≈ 8 × 109 | ❌ No |
| (B) | ~67 | 267 ≈ 1.5 × 1020 | ✅ Yes (approx) |
| (C) | 166 | 2166 ≈ 1050 | ❌ No |
| (D) | 100 | 2100 = 1.27 × 1030 | ✅ Exact (not listed) |
GATE Exam Relevance
- Microbiology PYQ tests exponential growth math
- Formula: n = t / g, N = 2n
- Approximation: 210 ≈ 103 simplifies large exponents
- Rich medium ensures log phase (no lag/stationary)
Final Answer: (B) (closest practical option to 2100)
