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embphysics

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## Homework Statement

Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0.What is A(0)?

## Homework Equations

## The Attempt at a Solution

The answer to this question is [itex]\displaystyle \frac{dA}{dt} = - \frac{1}{100} A(t)[/itex] Why is is so? Why is the differential equation proportional to the amount of salt at time t? Why couldn't to be, say, inversely proportional? What would the arguments be to show that it is in fact proportional, as opposed to any other alternatives?

I tried to reason through this myself. I said, suppose the rate at which the salt content changes is inversely proportional to the amount at time t. If initially there is a large amount of salt, then the rate at which the amount changes is initially small. Even though the rate is small, A(t) will still decrease, just slowly. So, as time passes, A(t) slowly diminishes. As A(t) decreases, the rate at which it continues to decrease becomes larger.

Through theses arguments I was hoping to show the unreasonableness of the differential equation being inversely proportional to the function. In spite of this, I am not really convinced.

I was hoping someone could provide me with some arguments as to why the differential equation is proportional to the function, as opposed to inversely, or proportional to the square, or the cube, etc.