# Acid Solution Tank Differential Equation: Determining Volume of Acid Over Time

• flash
In summary, the acid solution in a tank initially containing 200L of 0.5% acid solution is being constantly mixed and has a flow rate of 6L/min entering and 8L/min leaving. With the solution entering at a concentration of 20% acid, the volume of acid in the tank after t minutes can be determined using the equation \frac{dy}{dt} + \frac{8}{200-2t}y=1.2.
flash
An acid solution flows at a constant rate of 6L/min into a tank which initially holds 200L of a 0.5% acid solution. The solution in the tank is kept well mixed and flows out of the tank at 8L/min. If the solution entering the tank is 20% acid then determine the volume of acid in the tank after t minutes.

I just want to make sure I have the differential equation right.
let y(t) be the amount of acid solution in the tank.
rate in = 0.2 * 6 = 1.2L acid per min
rate out = y(t)/v(t) * 8, where v(t) = 200 - 2t is the volume of liquid in the tank

So I got

$$\frac{dy}{dt} + \frac{8}{200-2t}y=1.2$$

Can someone please confirm that this is the right equation to be working with?

Yes, that is correct.

## 1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes with respect to one or more independent variables. They involve derivatives, which represent how a quantity changes over time or space.

## 2. What are some real-world applications of differential equations?

Differential equations are used to model a wide range of phenomena in the physical, biological, and social sciences. They can be used to describe the motion of objects, the spread of disease, population growth, and many other systems.

## 3. How are differential equations solved?

There are several methods for solving differential equations, including separation of variables, substitution, and the use of integrating factors. Depending on the complexity of the equation, numerical methods may also be used to find approximate solutions.

## 4. What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve only one independent variable, while partial differential equations involve multiple independent variables. This means that solutions to partial differential equations can vary in multiple dimensions, whereas solutions to ordinary differential equations vary along a single dimension.

## 5. Why are differential equations important in science?

Differential equations are essential for understanding and predicting the behavior of systems in the natural and social sciences. They provide a powerful tool for modeling complex systems and making predictions about how they will change over time. Many scientific theories, such as Newton's laws of motion, are based on differential equations.

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