GATE 2022 BT Q.33: solution and answer

Official answer: 1Source: Official answer key, GATE 2022 BT (IIT Kharagpur)

1. In the exponential phase, biomass grows as $\frac{dX}{dt}=\mu X$. Here $X$ is the cell (biomass) concentration and $\mu$ is the specific growth rate in ${\text{h}}^{-1}$. Integrating from $X_0$ at $t=0$ gives $X=X_0{e}^{\mu t}$.

2. The doubling time $t_d$ is the time it takes for $X$ to reach $2X_0$. Putting that into the growth law gives $2=e^{\mu t_d}$. Taking the natural log of both sides: $$t_d=\frac{\ln 2}{\mu }⟺\mu =\frac{\ln 2}{t_d}$$

3. Substitute $\ln 2\approx 0.693$ and $t_d=0.693\text{h}$: $$\mu =\frac{0.693}{0.693\text{h}}=1{\text{h}}^{-1}$$ (Using $\ln 2=0.6931$ gives $1.0002{\text{h}}^{-1}$, which also rounds to 1.)

4. Rounded to the nearest integer, $\mu =1{\text{h}}^{-1}$.