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Question 58
Question Statement:
Mammalian cells in active growth phase were seeded at a density of 1105 cells/ml. After
72 hours, 1106 cells/ml were obtained. The population doubling time of the cells in hours
is (up to two decimal places) _________
Official Answer Key:
21.00 to 22.00
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2018 Q.58)**
**1. Governing Concept & Definition**:
- **Topic**: Animal Cell Culture & Population Doubling Time
- Cell proliferation follows:
$N(t) = N_0 \cdot 2^{t / t_d} \implies t_d = \frac{t \ln 2}{\ln(N(t)/N_0)}$
**2. Detailed Step-by-Step Calculation**:
- **Given**:
- Seeding density $N_0 = 1 \times 10^5\text{ cells/mL}$
- Final density $N(72) = 1 \times 10^6\text{ cells/mL}$ (10-fold increase)
- Time elapsed $t = 72\text{ hours}$
- **Doubling Time**:
$t_d = \frac{72 \times \ln 2}{\ln(10)} = \frac{72 \times 0.69315}{2.302585} = \frac{49.907}{2.302585} \approx 21.67\text{ hours}$
**3. Final Answer**:
- **Calculated Value**: **21.7 hours**
- **Official GATE Accepted Range**: **21.00 to 22.00**
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