Step 1: Write the Exponential Equation

We are given that:

• Current population, $a = 66,500$
• The population decreases by 4% each year, which means the decay rate $b = 1 - 0.04 = 0.96$.

The exponential equation that models the population $y$ after $x$ years is: $y = 66,500 \times (0.96)^x$

Step 2: Calculate the Population in 10 Years

Now, we substitute $x = 10$ into the equation to predict the population in 10 years: $y = 66,500 \times (0.96)^{10}$

Using a calculator: $y \approx 66,500 \times 0.665$ $y \approx 44,222.5$

Final Answer:

To the nearest hundred, Antonio can expect the population of his county to be around 44,200 people in 10 years.

Would you like more details or clarifications on any part of this? Here are five related questions to explore:

1. How would the equation change if the population were increasing by 4% instead?
2. What would the population be after 5 years instead of 10?
3. How long would it take for the population to decrease to half of its original size?
4. How does changing the percentage rate of decay affect the long-term population?
5. What would happen to the population if the decrease rate were reduced to 2%?

Tip: For exponential decay problems, always remember that the base $b$ is calculated as $1 - r$, where $r$ is the rate of decrease.