# Salt Tank Differential Equation

• Puk3s
In summary, the conversation discusses the concentration and amount of salt in a tank containing 100 kg of salt and 2000 L of water. The solution enters the tank at a rate of 5 L/min with a concentration of 0.025 kg/L, and drains at the same rate. The concentration in the tank initially is 0.05 kg/L and will approach 0.025 kg/L as time approaches infinity. To find the amount of salt in the tank after 4.5 hours, an equation can be set up using the rate of change of salt in the tank. However, the concentration of salt in the solution will depend on time.
Puk3s
Ok so this is the question

A tank contains 100 kg of salt and 2000 L of water. A solution of a concentration 0.025 kg of salt per liter enters a tank at the rate 5 L/min. The solution is mixed and drains from the tank at the same rate.

A.What is the concentration of our solution in the tank initially

B.Find the amount of salt in the tank after 4.5 hours

C.Find the concentration of salt in the solution in the tank as time approaches infinity.

I found A to be .05 and C to be .025. But I'm not sure how to get B... I know I have to set up an equation but I'm not very good at these word problems and could use some help setting up the equation and answering B so any help would be great!

Thanks!

Rate of change of salt in tank = amount of salt in/min - amount of salt out/min

For the amount of salt in, just plug in the values you are given

$$Concentration * flowrate = \frac{.025kg}{L}\left(\frac{5L}{min}\right)$$

For salt out, the concentration will depend on t. More explicitly, since concentration is
kg/L, and the number of liters will never change (since the rate of flow in and the rate of flow out are the same), the amount of salt will change with time.

## What is a Salt Tank Differential Equation?

A Salt Tank Differential Equation is a mathematical model used to describe the dynamics of a saltwater tank. It takes into account factors such as the volume of the tank, the concentration of salt, and the rate of inflow and outflow of water to predict the changes in salinity over time.

## Why is a Salt Tank Differential Equation important?

A Salt Tank Differential Equation is important because it allows us to understand and predict the behavior of a saltwater tank. This information is crucial for maintaining a healthy and stable environment for aquatic life and can also be used in the design and control of saltwater tanks for various purposes.

## What are the variables in a Salt Tank Differential Equation?

The variables in a Salt Tank Differential Equation include the volume of the tank, the concentration of salt, the flow rate of water in and out of the tank, and time. These variables can be used to calculate the rate of change of salinity in the tank.

## How is a Salt Tank Differential Equation solved?

A Salt Tank Differential Equation can be solved using various mathematical techniques such as separation of variables, substitution, or using software programs. The solution will depend on the specific parameters and initial conditions of the saltwater tank being modeled.

## What are some applications of Salt Tank Differential Equations?

Salt Tank Differential Equations have various applications in fields such as aquaculture, oceanography, and environmental science. They are also used in the design and control of desalination plants and in the study of marine ecosystems. Additionally, they can be used to model the behavior of other types of tanks or systems involving the mixing of substances.

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