Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
A tank initially contains 50 gallons of fresh water. Brine contains 2 pounds per gallon of salt, flows into the tank at the rate of 2 gallons per minutes and the mixture kept uniform by stirring, runs out at the same rate. If it will take for the quantity of slat in the tank to increase from 40 to 80 pounds (in seconds) is $206x$, then find $x$. (given $\ln 3 = 1.0986$)
Step-by-Step Solution
Key Concept: Rate of salt loss equals inflow rate minus outflow rate, leading to a first-order linear differential equation solved by separation of variables.
The salt content $u$ at time $t$ changes at rate $\frac{du}{dt} = 4 - 2c$, where $c = \frac{u}{50}$ is the concentration. This gives $\frac{du}{dt} = 4 - \frac{2u}{50}$. Separating variables and integrating: $t = -25\ln(100-u) + K$. Using initial condition $u=0$ at $t=0$ gives $K = 25\ln(100)$. When $u$ changes from $40$ to $80$ lb, we get $t_2 - t_1 = 25\ln\left(\frac{100}{20}\right) = 25\ln(5) \approx 40.24$ minutes.
Correct Answer: 8