JEE Main 2022 (Online) 30th June Morning ShiftQuestion 1
A hostel has $100$ students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on $4^{\text{th}}$ day is $30$, then number of infected students on $8^{\text{th}}$ day will be $\_\_\_\_\_\_$ .
Differential Equation
Answer$90$
Solution
Concept. Rate $\propto I(100-I)$ is the logistic equation. Formulas used. (1) $\dfrac{dI}{dt}=kI(100-I)\Rightarrow I=\dfrac{100}{1+Ae^{-100kt}}$Step 1 - Constant $A$. $I(0)=2$: $1+A=50$, so $A=49$. Step 2 - Day 4. Let $r=e^{-400k}$. $I(4)=30$: $1+49r=\frac{100}{30}=\frac{10}{3}$, so $r=\frac{1}{21}$. Step 3 - Day 8. $e^{-800k}=r^2$: $1+49r^2=1+\frac{49}{441}=\frac{10}{9}$. Step 4 - Value. $I(8)=\dfrac{100}{10/9}=90$. Answer: 90