Differential Equation (Water/Solution Tank) problem

• Fragster
In summary, a differential equation is a mathematical equation used to relate the rate of change of a quantity to its current value. In the context of a water/solution tank problem, it is applied to model the change in volume of the liquid in the tank over time. The equation is set equal to the inflow and outflow rates and solved to determine the volume of liquid at any given time. The key variables in this type of problem include the volume of the tank, inflow and outflow rates, and initial volume of liquid. Common assumptions include a constant inflow and outflow rate, uniform concentration, and no other influencing factors. This type of problem has various real-world applications, such as predicting water levels, chemical reactions, and
Fragster

Homework Statement

A tank initially contains 200 gallons of fresh water, but then a salt solution of unknown concentration lb/gal is poured into the tank at 2 gal/min. The well-stirred mixture flows out of the tank at the same rate. After 120 minutes, the concentration of salt in the tank is 1.4 lb/gal. What is the concentration (in lb/gal) of the entering brine?

Homework Equations

Net rate of salt solution = rate of incoming salt solution - rate of outgoing salt solution

The Attempt at a Solution

I actually have the solution
280/(200-200(e^(-6/5))) or roughly, 2

but I cannot remember for the life of me how I got it :/

I think the biggest problem I have with the problem is the relationship between the rate of salt going in (x) and the net rate of salt.

Hi there,

First, let's define some variables:
- x = concentration of the entering brine (in lb/gal)
- y = rate of incoming salt solution (in lb/min)
- z = rate of outgoing salt solution (in lb/min)

We know that the net rate of salt solution is equal to the rate of incoming salt solution minus the rate of outgoing salt solution. So we can write the following equation:

y - z = 1.4 lb/gal * 2 gal/min

Next, we need to find the relationship between x, y, and z. We know that the concentration of salt in the tank is increasing at a rate of 1.4 lb/gal every 2 minutes. This means that for every 2 minutes, 1.4 lb of salt is being added to the tank. Since we know that the tank initially contained 200 gallons of fresh water, we can write the following equation:

x * 2 gal/min * 120 min = 200 gal * 0 lb/gal + 1.4 lb/gal * 2 gal/min * 120 min

Solving for x, we get:

x = 0.014 lb/gal

Therefore, the concentration of the entering brine is 0.014 lb/gal.

I hope this helps! Let me know if you have any further questions or if you would like me to explain any part of the solution in more detail.

1. What is a differential equation in the context of a water/solution tank problem?

A differential equation is a mathematical equation that relates the rate of change of a quantity to its current value. In the context of a water/solution tank problem, it is used to model the change in volume of the liquid in the tank over time.

2. How is a differential equation applied to solve a water/solution tank problem?

To solve a water/solution tank problem using a differential equation, the rate of change of the volume of liquid in the tank is set equal to the inflow and outflow rates. This equation is then solved to determine the volume of liquid at any given time.

3. What are the key variables in a water/solution tank differential equation problem?

The key variables in a water/solution tank differential equation problem include the volume of the tank, the inflow and outflow rates, and the initial volume of liquid in the tank. These variables are used to set up the differential equation and solve for the volume of liquid at any given time.

4. What are the assumptions made when solving a water/solution tank differential equation problem?

Common assumptions made when solving a water/solution tank differential equation problem include assuming a constant inflow and outflow rate, a uniform concentration of the solution in the tank, and no other factors affecting the volume of liquid in the tank (such as evaporation).

5. How can a water/solution tank differential equation problem be used in real-world applications?

Differential equations in the context of a water/solution tank problem can be used in various real-world applications such as modeling water levels in reservoirs, chemical reactions in industrial tanks, and drug concentrations in the human body. They can also be used to optimize and control these systems by predicting changes in volume and concentration over time.

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