Modeling a tank with dyed water

Lesson 2M

Many first order differential equations follow the pattern given by the tank problem given below. You have a quantity of something–in this case dye, but it could also be bacteria or money or salt or something else–and you need to model how that quantity changes over time.

You set up the differential equation by calculating the sum of all the ways the quantity could increase, and subtracting the sum of all the ways that it could decrease.

The problem

You have a 300 L tank full of pure water. You also have some dyed water that contains 5 g of red dye per liter of water. You add this dyed water to the pure water at a rate of 2 L per second, while draining the tank at the same rate so that the amount of solution remains constant.

Find a formula for the amount of dye in the tank at any given time.

The solution

After you have thought about it for a bit, watch the video.