# 1st order DE - word problem

• ranger
In summary, the conversation discusses the amount of salt in a 120gal tank initially containing 90lbs of salt dissolved in 90gal of water. Saltwater with a concentration of 2lb/gal flows into the tank at a rate of 4gal/min, and the mixture flows out at a rate of 3gal/min. To find the amount of salt in the tank when it is full, a differential equation is used to determine the rate of change of salt in the tank. The rate of entering is found by multiplying the concentration of salt in the entering water by the rate at which it enters, and the rate of leaving is calculated the same way using the concentration of the leaving water, which is equal to the concentration of

#### ranger

Gold Member
A 120gal tank initially contains 90lbs of salt dissolved in 90gal of water. Saltwater containing 2lb/gal of salt flows into the tank at the rate of 4gal/min, and the well stirred mixture flows out of the tank at the rate of 3gal/min. How much salt does the tank contain when it is full?

Well here is my train of thought. Since they are asking for the amount of salt when the tank is full, it seems logical to model the differential equation with the amount of salt and time, since we know the rate of entering and rate of leaving, we can find how much time 30gals will take to accumulate.

So I will let Q be the quantity of salt. It seems that the rate of change of salt in the tank should be the difference between the rate of entering and rate of leaving, correct? Now If I'm wrong about this, then don't bother reading the rest, because I entire approach is based on this assumption.

I get the equation is this form:
$$\frac{dQ}{dt} = rate_{entering} - rate_{leaving}$$

It seems logical to have the rate of entering and leaving as lb/min since dQ/dt is also lb/min.

Rate of entering: 2lb/gal * 4gal/min = 8lb/min

Rate of leaving: this one is giving me some problems.

--thanks.

To find the rate entering, you multiplied the concentration of salt in the entering water by the rate at which the entering water is entering. So the rate of leaving is calculated the same way. You know that the rate at which the leaving water is leaving, all you need to know is the concentration of the leaving water. Since the tank is well-stirred, the concentration of the leaving water at time t is precisely the concenctration of the tank at time t, which would be Q(t)/A(t) where A(t) is the amount of water at time t. It's easy to find an expression for A(t).

I think I figured it out.
Rate of leaving = 3Q/(90+t)

correct?

## 1. What is a "1st order DE - word problem"?

A "1st order DE - word problem" refers to a first-order differential equation that is presented in the form of a real-world word problem. These types of problems involve finding a function that represents the relationship between a variable and its rate of change, given a set of initial conditions.

## 2. How do I solve a "1st order DE - word problem"?

To solve a "1st order DE - word problem," you will need to use techniques such as separation of variables, integrating factors, or substitution, depending on the specific problem. The goal is to manipulate the equation until you can isolate the dependent variable on one side and the independent variable and its derivatives on the other side. Then, you can solve for the dependent variable using standard integration techniques.

## 3. What are the key steps to solving a "1st order DE - word problem"?

The key steps to solving a "1st order DE - word problem" are: 1) identifying the independent and dependent variables, 2) writing the equation in the standard form, 3) determining the necessary technique to solve the equation, 4) applying the technique to manipulate the equation, and 5) solving for the dependent variable and checking the solution with the initial conditions given in the problem.

## 4. Can you provide an example of a "1st order DE - word problem"?

Yes, here is an example: "A bacteria culture initially contains 1000 bacteria and grows at a rate of 50% per hour. How many bacteria will be present after 5 hours?" The equation for this problem is dN/dt = 0.5N, where N is the number of bacteria and t is time. By solving this first-order differential equation, we can determine that after 5 hours, there will be approximately 4365 bacteria present in the culture.

## 5. What are some real-world applications of "1st order DE - word problems"?

There are many real-world applications of "1st order DE - word problems," including population growth, radioactive decay, and drug dosage calculations. These types of problems are also commonly seen in fields such as physics, biology, and economics, where there is a need to model and understand the relationship between variables and their rates of change.