A 500-gallon tank initially contains 400 gallons of brine containing 75 pounds of dissolved salt. Brine containing 3 pounds of salt per gallon flows into the tank at the rate of 4 gallons per minute, and the well-stirred mixture flows out of the tank at the rate of 1 gallon per minute. Set up a differential equation for the amount of salt, A(t), in the tank at time t

To set up a differential equation for the amount of salt \( A(t) \) in the tank at time \( t \), we need to account for both the inflow and outflow of brine:

### Given:
- Initial amount of brine: \( 400 \) gallons
- Initial amount of salt: \( 75 \) pounds
- Inflow rate: \( 4 \) gallons per minute
- Salt concentration of inflow: \( 3 \) pounds per gallon
- Outflow rate: \( 1 \) gallon per minute

### Step 1: Inflow of Salt
The brine enters the tank at a rate of \( 4 \) gallons per minute with a concentration of \( 3 \) pounds per gallon. Thus, the rate at which salt enters the tank is:
\[
\text{Rate of salt inflow} = 4 \, \text{gallons per minute} \times 3 \, \text{pounds per gallon} = 12 \, \text{pounds per minute}
\]

### Step 2: Outflow of Salt
The brine flows out at a rate of \( 1 \) gallon per minute. Since the tank is well-stirred, the concentration of salt in the outgoing brine will be the current concentration in the tank:
\[
\text{Concentration of salt in the tank} = \frac{A(t)}{400 + 3t}
\]
where \( 400 + 3t \) represents the total volume of the brine in the tank at time \( t \), accounting for the net increase of \( 3 \) gallons per minute (since \( 4 \) gallons are flowing in and \( 1 \) gallon is flowing out).

Thus, the rate at which salt flows out is:
\[
\text{Rate of salt outflow} = 1 \, \text{gallon per minute} \times \frac{A(t)}{400 + 3t} = \frac{A(t)}{400 + 3t}
\]

### Step 3: Differential Equation
The rate of change of the amount of salt in the tank is given by the difference between the rate of salt inflow and the rate of salt outflow:
\[
\frac{dA}{dt} = 12 - \frac{A(t)}{400 + 3t}
\]

This equation describes the amount of salt \( A(t) \) in the tank at any time \( t \).

Would you like a detailed solution or have further questions? Here are some related problems:

1. How would the equation change if the inflow concentration was different?
2. What would happen if the outflow rate was increased to 2 gallons per minute?
3. How long would it take for the tank to reach 500 gallons?
4. What would be the concentration of salt in the tank after 10 minutes?
5. How can this differential equation be solved analytically?

**Tip:** When setting up differential equations, carefully account for changes in volume and concentration over time.

A 500-gallon tank initially contains 400 gallons of brine containing 75 pounds of dissolved salt. Brine containing 3 pounds of salt per gallon flows into the tank at the rate of 4 gallons per minute, and the well-stirred mixture flows out of the tank at the rate of 1 gallon per minute. Set up a differential equation for the amount of salt, A(t), in the tank at time t.

Learn how to set up a differential equation to model the amount of salt in a brine tank. This problem involves rates of change, volume flow, and concentration, using differential equations. Suitable for Grades 11-12 and introductory college courses.

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