Question

Antonio looked up the most recent census data for his county. He learned that the current population is about 66,500 people, and the population is expected to decrease by 4% each year. Write an exponential equation in the form y=a(b)x that can model the population of his county, y, in x years. Use whole numbers, decimals, or simplified fractions for the values of a and b. y = To the nearest hundred people, what can Antonio expect the population of his county to be in 10 years? people

Answer 1

The exponential decay model for Antonio's county population is y = 66,500(0.96)^x. After 10 years, using this model, the population is expected to be approximately 44,200 people, to the nearest hundred.

Step-by-step explanation:

To write an exponential equation in the form y = a(b)^x that models the population of Antonio's county, where y is the population in x years, we start with the current population as the initial value a.

Here the initial population is 66,500. Since the population decreases by 4% each year, the growth factor b would be 1 - 0.04 = 0.96. The exponential decay model for the population then becomes y = 66,500(0.96)^x.

To find the expected population to the nearest hundred people after 10 years, we substitute x = 10 into the equation: y = 66,500(0.96)^10.

When this is calculated, we get y = 66,500(0.6648), which equals approximately 44,199 to the nearest hundred. Therefore, Antonio can expect the population of his county to be around 44,200 people in 10 years.