Question Statement:
Consider the equation
2
aS
V
S
b
S
c
Given a = 4, b = 1 and c = 9, the positive value of S at which V is maximum, will be _______.
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2016 Q.65)**
**1. Governing Concept & Definition**:
- **Topic**: Enzyme Kinetics & Substrate Inhibition
- For substrate-inhibited enzyme kinetics (Haldane model):
$v = \frac{a S}{S^2 + b S + c}$
The maximum velocity occurs at the optimal substrate concentration $S_{\text{opt}} = \sqrt{c}$.
**2. Detailed Step-by-Step Calculation**:
- **Given Parameters**:
- $a = 4$
- $b = 1$
- $c = 9$
- **Differentiating to find maximum**:
$\frac{dv}{dS} = \frac{a(S^2 + b S + c) - a S(2S + b)}{(S^2 + b S + c)^2} = \frac{a(c - S^2)}{(S^2 + b S + c)^2} = 0$
$c - S^2 = 0 \implies S^2 = c = 9 \implies S = \sqrt{9} = 3.0$
**3. Final Answer**:
- **Calculated Value**: **3.0**
- **Official GATE Accepted Range**: **3.0 to 3.0**
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