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Related Rate Applications

In a Nut Shell: There are a number of related rated applications from a general family

of linear, first-order differential equations that are either separable or where an

integrating factor can be used to obtain a solution. One common application is the

changing concentration of a solute in a tank with inflow and outflow. Another is the

increase in a savings account related to deposits. First consider flow into and exiting a

tank as shown in the figure below.

Applications involving changes of solute in a Tank

Basic Differential Equation governing Related Rate for inflow and outflow

dy/dt = Rate of inflow ˗ Rate of outflow = rate of change of solute

of solute of solute in the tank

y(t) = amount of solute in the tank at any time t

dy/dt = rate of change of solute in the tank at any time t

V(t) = volume of substance within tank at any time t

where:

Rate of inflow = r_i C_i , where r_i = rate of flow in i.e. Liters per minute

C_i = concentration of solute entering in grams/liter

Rate of outflow = r_o C_o , where r_o = rate of flow out i.e. Liters per minute

C_o = concentration of solute exiting in grams/liter

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