Differential Equations
Differential Equations
Allen Star Batch
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: Set up a first-order linear differential equation modeling salt accumulation: du/dt = (inflow rate × salt concentration) - (outflow rate × current concentration). The solution requires separating variables and integrating to find time as a function of salt quantity: t = -25ln(100-u) + 25ln(100), then evaluate between u=40 and u=80 using logarithm properties.
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