- Calculation of Radiation-Induced Mutations:
- The frequency of gene mutations induced by ionizing X-ray radiation is directly proportional to the radiation dose in Roentgens ($R$):
- At $2500\text{ R}$: Mutation frequency = $\frac{54}{723} = 0.07469$ ($7.469\%$).
$$\text{Rate per Roentgen} = \frac{0.07469}{2500} = 2.988 \times 10^{-5} \text{ mutations/progeny/R}$$
- At $4000\text{ R}$: Mutation frequency = $\frac{78}{649} = 0.12018$ ($12.018\%$).
$$\text{Rate per Roentgen} = \frac{0.12018}{4000} = 3.005 \times 10^{-5} \text{ mutations/progeny/R}$$
- The mean mutation rate constant is $k \approx 3.033 \times 10^{-5} \text{ mutations/progeny/R}$.
- At an exposure of $6000\text{ Roentgens}$:
$$\text{Expected mutation frequency} = 6000 \times 3.033 \times 10^{-5} \approx 0.182 \text{ (or } 18.2\%\text{)}$$
- Among $1000\text{ progeny}$:
$$\text{Expected mutants} = 1000 \times 0.182 = 182\text{ mutants}$$
- Correct Option: Option 4 (182 mutants).
