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The problem states that the initial number of ants in a house is 64,000, and they decrease at a rate of 8% per hour. We need to determine which equation represents the number of ants remaining after \( x \) hours.

### Steps to solve:

1. **Initial number of ants**: \( 64,000 \).
2. **Rate of decrease**: 8% per hour means the ants decrease by \( 8\% \), or \( 0.08 \), each hour. The remaining percentage of ants after one hour is \( 100\% - 8\% = 92\% \), or \( 0.92 \).
3. **Exponential decay formula**: The general form of an exponential decay function is:
   \[
   y = A \cdot (r)^x
   \]
   Where:
   - \( y \) is the amount remaining after \( x \) hours.
   - \( A \) is the initial amount (64,000).
   - \( r \) is the remaining percentage (in this case \( 0.92 \)).

4. Plugging in the values, the equation becomes:
   \[
   y = 64000 \cdot (0.92)^x
   \]

So, the correct answer is **C. \( y = 64000(0.92)^x \)**.

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Would you like more details or have any questions? Here are some related questions to further explore this concept:

1. What is the equation for growth instead of decay?
2. How does changing the decay rate affect the equation?
3. How would you modify the equation if the ants decreased by 10% per hour?
4. Can exponential decay ever result in zero ants remaining?
5. What is the difference between exponential decay and linear decay?

**Tip**: Always remember that a decay rate less than 1 indicates a decrease, while a rate greater than 1 indicates growth!

An exterminator estimates that a house has approximately 64,000 ants. After treating the house, the number of ants decreases at a rate of 8% per hour. Which equation represents this situation where y is the number of ants remaining after x hours?

Learn how to represent the situation where 64,000 ants decrease at a rate of 8% per hour using the exponential decay equation y = 64000 * (0.92)^x. This problem covers key algebraic concepts of exponential decay, suitable for students in Grades 6-8.

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