Initially, 15 grams of salt are dissolved into 32 liters of water. Brine with a concentration of 3 grams per liter is added at a rate of 2 liters per minute. The tank is well mixed and drained at 2 liters per minute. Let x be the amount of salt in grams, in the solution after t minutes have lapsed. Find a formula for the incremental change in the salt, delta x, in terms of the amount of salt in the solution x and the incremental change in time delta t. delta x = _______ grams. 

We are given a problem involving a brine tank where salt is added and removed at the same rate of 2 liters per minute, and the goal is to find a formula for the incremental change in salt, \( \Delta x \), in terms of the amount of salt in the tank \( x \) and the incremental time change \( \Delta t \).

### Step-by-Step Breakdown:

1. **Initial Conditions:**
   - Initially, there are 15 grams of salt in the 32-liter solution.
   - Brine with a concentration of 3 grams per liter is added at a rate of 2 liters per minute.
   - The solution is drained at the same rate of 2 liters per minute.

2. **Amount of Salt Added:**
   The incoming brine has a concentration of 3 grams per liter, and it flows in at a rate of 2 liters per minute. Therefore, the amount of salt entering the tank per minute is:
   \[
   \text{Salt In} = (3 \, \text{grams/liter}) \times (2 \, \text{liters/minute}) = 6 \, \text{grams/minute}
   \]

3. **Amount of Salt Drained:**
   The solution is drained at a rate of 2 liters per minute. The concentration of salt in the tank at any time is given by:
   \[
   \text{Concentration in Tank} = \frac{x(t)}{32} \, \text{grams/liter}
   \]
   So, the amount of salt being removed per minute is:
   \[
   \text{Salt Out} = \left( \frac{x(t)}{32} \, \text{grams/liter} \right) \times (2 \, \text{liters/minute}) = \frac{2x(t)}{32} = \frac{x(t)}{16} \, \text{grams/minute}
   \]

4. **Rate of Change of Salt:**
   The rate of change of the amount of salt in the tank is the difference between the rate at which salt is added and the rate at which it is drained:
   \[
   \frac{dx}{dt} = \text{Salt In} - \text{Salt Out} = 6 - \frac{x(t)}{16}
   \]

5. **Incremental Change in Salt, \( \Delta x \):**
   Over a small time interval \( \Delta t \), the change in the amount of salt, \( \Delta x \), can be approximated as:
   \[
   \Delta x = \left( 6 - \frac{x}{16} \right) \Delta t
   \]

Thus, the formula for the incremental change in salt is:

\[
\Delta x = \left( 6 - \frac{x}{16} \right) \Delta t \, \text{grams}.
\]

Would you like more details or have any questions?

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### Related Questions:
1. How do we find the general solution for \( x(t) \) from the differential equation \( \frac{dx}{dt} = 6 - \frac{x}{16} \)?
2. What happens to the concentration of salt after a long period of time?
3. How would the formula change if the brine concentration added were different?
4. What is the equilibrium amount of salt in the tank?
5. How long does it take for the amount of salt in the tank to reach half of its equilibrium value?

**Tip:** Understanding how salt is added and removed from a system like this can help when analyzing similar mixing problems in chemistry or engineering.

Initially, 15 grams of salt are dissolved into 32 liters of water. Brine with a concentration of 3 grams per liter is added at a rate of 2 liters per minute. The tank is well mixed and drained at 2 liters per minute. Let x be the amount of salt in grams, in the solution after t minutes have lapsed. Find a formula for the incremental change in the salt, delta x, in terms of the amount of salt in the solution x and the incremental change in time delta t. delta x = _______ grams.

Solve the incremental change in the amount of salt in a well-mixed brine tank where salt is added and drained at equal rates. Learn to calculate Δx in terms of the amount of salt and time, using first-order linear differential equations.

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