1. ## modeling with diffEQs

I have been fueling an aneurysm trying to figure these problems out. I just cant put together a formula for a single modeling question. My professor never really went over the process of how to write an equation for them, he more or less assumed we could read and that would suffice for the class.

For instance:
"Consider a tank used in experiments. After 1 experiment the tank contains 200 liters of a dye solution with a concentration of 1 g/liter. To prepare for the next experiment the tank is to be rinsed with fresh water flowing in at a rate of 2 liters/min, and the well-stirred solution is flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value."

I know I have to find a differential equation for the scenario, then I have to solve it for t (time) and find out what value t is when the equation is (0.01) of the original 100 g/liter. I just don't know how to write it out. Any help you could possibly give on the process of tackling modeling questions would be super awesome and also fantastic.

Thanks.

2. Originally Posted by DinoJockey
I have been fueling an aneurysm trying to figure these problems out. I just cant put together a formula for a single modeling question. My professor never really went over the process of how to write an equation for them, he more or less assumed we could read and that would suffice for the class.

For instance:
"Consider a tank used in experiments. After 1 experiment the tank contains 200 liters of a dye solution with a concentration of 1 g/liter. To prepare for the next experiment the tank is to be rinsed with fresh water flowing in at a rate of 2 liters/min, and the well-stirred solution is flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value."

I know I have to find a differential equation for the scenario, then I have to solve it for t (time) and find out what value t is when the equation is (0.01) of the original 100 g/liter. I just don't know how to write it out. Any help you could possibly give on the process of tackling modeling questions would be super awesome and also fantastic.

Thanks.
Let y(t) be the amount of dye in the tank, in grams, at time t, in minutes. Then dy/dt is the rate at which dye is flowing from the tank. Since there are always 200 liters of water in the tank, the concentration is y/200 g/l and since water is flowing out at 2 l/min, it carries (2 l/m)(y/200 g/l)= y/100 g/m with it.
$\displaystyle \frac{dy}{dt}= -y/100$
with initial condition [tex]y(0)= (0.1 g/l)(200 l)= 20 g.

Solve that differential equation for y(t) (it's a fairly simple exponential, set it equal to (0.1 g/l)(0.01)(200 l)= .2 g, and solve for t.