Mathematics

Mathematics

Problem

A 200-liter tank initially contains 50 grams of salt uniformly dissolved. Pure water flows in at 5 L/min, and the well-mixed solution is pumped out at 4 L/min. Determine a differential equation for y(t)y(t), the amount of salt (in grams) in the tank after tt minutes.
Intuitive explanation

The key is to track salt using a rate-in minus rate-out principle. Salt enters the tank only through the outflow (since pure water comes in), and leaves through the outflow at a concentration proportional to how much salt is currently dissolved. The volume changes over time because more water enters than leaves, so the concentration y(t)/V(t)y(t)/V(t) depends on the growing tank volume. Set up dydt\frac{dy}{dt} by calculating the salt lost per minute through the pump.

Intuitive explanation

The key is to track salt using a rate-in minus rate-out principle. Salt enters the tank only through the outflow (since pure water comes in), and leaves through the outflow at a concentration proportional to how much salt is currently dissolved. The volume changes over time because more water enters than leaves, so the concentration y(t)/V(t)y(t)/V(t) depends on the growing tank volume. Set up dydt\frac{dy}{dt} by calculating the salt lost per minute through the pump.

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