# When is the tank empty? [differential equation]

In summary, the conversation discussed two problems involving a large tank filled with 500 gallons of pure water being pumped with brine containing 2 lbs of salt per gallon at a rate of 5 gal/min. The solution is then pumped out at the same rate. The first problem asked for the number of lbs of salt in the tank at a given time, while the second problem asked for the time when the tank would be empty assuming the solution is pumped out at a faster rate of 10 gal/min. It was clarified that the concentration of salt is not relevant to the second problem and that the solution involves two parts: finding the salt content as a function of time and determining when the tank will be empty.

## Homework Statement

problem 23:

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 lbs of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well mixed solution is pumped out at the same rate. Find the number A(t) of lbs of salt in the tank at time t.

problem 25: Solve problem 23 under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty?

I'm only doing problem 25 here. I've already done problem 23.

A(0) = 500

## The Attempt at a Solution

I really don't know where to start. Discussing with someone, they said that the "brine" is important for this problem, but I really do not understand how.

It is asking for when the tank is empty. I don't see how the concentration of salt has any importance here. I don't see how the amount of salt that is present in a diluted amount of solution changes how much solution there is.

I have $\frac {dA}{dt} = R_{in} - R_{out}$

So at time t=0, 500 gallons of water are present

$5 \frac {gal}{min}$ of water, saturated heavily with salt at 2lbs per gallon is being pumped in but this isn't even relevant to this problem because its only asking about when water in the tank is 0, not about a concentration.

After t minutes, since the incoming amount of water is 5 gallons/min, and the water coming out is 10 gallons/min, you have -5 gallons per minute coming out.

At this point I'm lost. Since I'm trying to make an equation here, and 5 gallons/min is already in units, if I try to add a variable t like 5t, it makes the units just 5 gallons, so that's no good.

Can someone please not spoil the problem, but just confirm to me that the concentration of salt really doesn't matter? Or if it does, can anyone explain why, because I really don't see it right now. In the mean time I'm going to try to figure out how I can make an equation out of this.

Yellowkies_3275

## Homework Statement

problem 23:

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 lbs of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well mixed solution is pumped out at the same rate. Find the number A(t) of lbs of salt in the tank at time t.

problem 25: Solve problem 23 under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty?

I'm only doing problem 25 here. I've already done problem 23.

A(0) = 500

## The Attempt at a Solution

I really don't know where to start. Discussing with someone, they said that the "brine" is important for this problem, but I really do not understand how.

It is asking for when the tank is empty. I don't see how the concentration of salt has any importance here. I don't see how the amount of salt that is present in a diluted amount of solution changes how much solution there is.

I have $\frac {dA}{dt} = R_{in} - R_{out}$

So at time t=0, 500 gallons of water are present

$5 \frac {gal}{min}$ of water, saturated heavily with salt at 2lbs per gallon is being pumped in but this isn't even relevant to this problem because its only asking about when water in the tank is 0, not about a concentration.

After t minutes, since the incoming amount of water is 5 gallons/min, and the water coming out is 10 gallons/min, you have -5 gallons per minute coming out.

At this point I'm lost. Since I'm trying to make an equation here, and 5 gallons/min is already in units, if I try to add a variable t like 5t, it makes the units just 5 gallons, so that's no good.

Can someone please not spoil the problem, but just confirm to me that the concentration of salt really doesn't matter? Or if it does, can anyone explain why, because I really don't see it right now. In the mean time I'm going to try to figure out how I can make an equation out of this.

It looks to me as though problem 25 has two parts. Part (i): what is salt content A(t) as a function of t?; and (ii) when is the tank empty?

Certainly, (ii) has nothing to do with (i), but if you read what is actually written it said to solve problem 23 in the new setup, and that problem asked for A(t).

Ray Vickson said:
It looks to me as though problem 25 has two parts. Part (i): what is salt content A(t) as a function of t?; and (ii) when is the tank empty?

Certainly, (ii) has nothing to do with (i), but if you read what is actually written it said to solve problem 23 in the new setup, and that problem asked for A(t).

Gosh you are a life saver. I don't know how I missed the "solve problem 23" part even though I clearly wrote it out.

Will report back in a minute.

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a quantity and its rate of change. It is commonly used in physics, engineering, and other scientific fields to model and predict the behavior of complex systems.

## 2. How is a differential equation used to determine when a tank is empty?

A differential equation can be used to model the rate of change of the volume of liquid in a tank. By setting the rate of change equal to zero, we can solve for the time at which the tank will be empty.

## 3. Can a differential equation account for external factors that may affect the rate of change?

Yes, a differential equation can be modified to include external factors such as inflow or outflow rates, evaporation, or changes in temperature that may affect the rate of change of the liquid in the tank.

## 4. Are there different types of differential equations that can be used to model a tank being emptied?

Yes, there are various types of differential equations that can be used, depending on the specific characteristics of the tank and the liquid being emptied. These include linear and non-linear differential equations, as well as ordinary and partial differential equations.

## 5. Can a differential equation accurately predict when a tank will be empty?

Yes, a properly constructed and solved differential equation can provide an accurate prediction of when a tank will be empty. However, the accuracy of the prediction may be affected by external factors and the assumptions made in the modeling process.

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