A desalinization process is eliminating 10% of the salt from seawater every 10 minutes. There are initially 350 g of salt in 10 liters of water. Find an exponential function for *A*, the grams of salt in the 10 liters of seawater *t* minutes after the process has begun. (See Example 5 in this section.)

To solve this problem, we need to establish an exponential function that models the amount of salt remaining in the seawater after \( t \) minutes, given that 10% of the salt is eliminated every 10 minutes.

### Step-by-Step Explanation:

1. **Initial Condition**:
   - Initially, there are 350 grams of salt in the seawater.
   - This is the starting value of the salt, \( A(0) = 350 \).

2. **Decay Rate**:
   - Every 10 minutes, 10% of the salt is eliminated, which means 90% of the salt remains. Therefore, the decay factor for every 10 minutes is \( 0.9 \).

3. **Time Factor**:
   - The decay factor \( 0.9 \) applies every 10 minutes. If \( t \) is measured in minutes, we need to divide \( t \) by 10 to determine how many 10-minute intervals have passed.

4. **Exponential Function**:
   - The general form of the exponential decay function is:
     \[
     A(t) = A_0 \cdot e^{kt}
     \]
     But since we already know the decay rate every 10 minutes, we can write:
     \[
     A(t) = A_0 \cdot (0.9)^{t/10}
     \]
     where:
     - \( A_0 = 350 \) grams (initial amount of salt),
     - \( 0.9 \) is the fraction of salt remaining after each 10-minute interval,
     - \( \frac{t}{10} \) represents the number of 10-minute intervals.

Thus, the exponential function for \( A \), the amount of salt in grams after \( t \) minutes, is:

\[
A(t) = 350 \cdot (0.9)^{t/10}
\]

### Interpretation:
- \( A(t) \) gives the amount of salt left after \( t \) minutes.
- As \( t \) increases, \( A(t) \) decreases exponentially because 10% of the salt is being eliminated every 10 minutes.

Would you like further details on this, or do you have any questions?

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Here are 5 related questions you might find helpful:

1. How long will it take for the salt content to be reduced to half its original amount?
2. What would the function look like if 20% of the salt were removed every 10 minutes instead of 10%?
3. How much salt is left after 30 minutes of the desalination process?
4. If the initial amount of salt were 500 g, how would the function change?
5. What is the rate of change of salt content at any given time?

**Tip**: Exponential decay models are commonly used in processes like radioactive decay and cooling, where a percentage of a quantity decreases over fixed time intervals.

A desalinization process is eliminating 10% of the salt from seawater every 10 minutes. There are initially 350 g of salt in 10 liters of water. Find an exponential function for A, the grams of salt in the 10 liters of seawater t minutes after the process has begun.

Learn how to model the amount of salt left after a desalination process removes 10% of the salt every 10 minutes. The initial condition involves 350 grams of salt in 10 liters of water. This problem uses an exponential decay function and is suitable for Grades 10-12.

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