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This problem involves a mixture of salt and water flowing in and out of a tank, and we need to calculate the amount of salt in the tank at a particular time.

Let's break it down step by step.

### Information:
- The tank starts with **100 gallons** of fresh water.
- Saltwater containing **1/2 lb of salt per gallon** flows into the tank at a rate of **2 gallons per minute**.
- The mixture flows out at the same rate of **2 gallons per minute** for the first 10 minutes.
- After 10 minutes, the inflow is switched to fresh water at a rate of **1 gallon per minute**, and the outflow remains the same at **1 gallon per minute** for an additional 10 minutes.

### Step 1: Set up the initial conditions for the first 10 minutes.

Let:
- \( S(t) \) represent the amount of salt in the tank at time \( t \), where \( t \) is in minutes.
- The inflow rate of salt is **\( \frac{1}{2} \times 2 = 1 \, \text{lb/min} \)**.
- The outflow rate depends on the concentration of salt in the tank, which is \( \frac{S(t)}{100} \, \text{lb/gal} \), multiplied by the outflow rate of **2 gal/min**.

The differential equation for the salt in the tank for the first 10 minutes is:

\[
\frac{dS}{dt} = 1 - \frac{S(t)}{100} \times 2
\]

This simplifies to:

\[
\frac{dS}{dt} = 1 - \frac{2S(t)}{100} = 1 - \frac{S(t)}{50}
\]

This is a separable differential equation.

### Step 2: Solve the differential equation for the first 10 minutes.

Rearranging:

\[
\frac{dS}{1 - \frac{S(t)}{50}} = dt
\]

We can solve this equation and find \( S(t) \) over the first 10 minutes.

### Step 3: Set up the equation for the next 10 minutes.

After 10 minutes, the salt inflow stops (now fresh water flows in), so the inflow rate of salt becomes **0 lb/min**. The outflow rate of the mixture is now **1 gallon per minute**. The differential equation for this stage becomes:

\[
\frac{dS}{dt} = - \frac{S(t)}{100}
\]

Again, this is a separable equation that can be solved for the amount of salt at the end of the additional 10 minutes.

Let me calculate these stages step by step to find the final amount of salt after the 20 minutes.It looks like an issue arose when solving for the constant during the second stage. Let me correct this by carefully re-evaluating the constants for the second differential equation, ensuring that the solution for \(S(t)\) at \(t = 10\) smoothly transitions to the second phase of the problem. I will adjust and proceed with calculations for the final salt content after 20 minutes.The amount of salt in the tank after 20 minutes is approximately **8.20 pounds**, rounded to two decimal places.

### Recap of the process:
- **First 10 minutes**: Saltwater is entering the tank at 2 gal/min, contributing 1 lb of salt per minute. The amount of salt in the tank after 10 minutes is calculated based on this inflow and simultaneous outflow.
- **Next 10 minutes**: Freshwater flows into the tank (no salt inflow), but salt continues to leave, reducing the overall amount of salt.

Would you like further details or clarifications on any part of the process?

### Follow-up questions:
1. How would the amount of salt change if the inflow concentration was different?
2. What if the outflow rate was higher than the inflow rate from the beginning?
3. How would the salt content be affected if there were no outflow in the first 10 minutes?
4. Can this problem be solved using numerical methods? How would you set it up?
5. How does the concept of differential equations apply to other similar real-world systems?

#### Tip:
In problems involving rates of change and mixing, it's often helpful to consider the inflow and outflow dynamics carefully, as they significantly affect the overall concentration and the differential equation used.

A tank originally contains 100 gal of fresh water. Then water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of 1 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. Round your answer to two decimal places.

This problem involves calculating the amount of salt in a tank over time using differential equations. The solution explains how to set up and solve the separable differential equations for both inflow and outflow of salt, and determine the final amount of salt in the tank after 20 minutes. Suitable for college-level students studying calculus or differential equations.

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