Question

1. the mayor of a city records the population each year since 1992. she models the data as $p(t) = 14.3(0.89)^t$ where p represents the city’s population, in thousands of people, and t represents the number of years since 1992. select all the statements that are true based on this model.

a) the population has increased by 11% each year.
b) the population has decreased by 11% each year.
c) the population was 89,000 people in 1992.
d) the population was 14,300 people in 1992.

Explanation:

Step1: Analyze the exponential model \( P(t) = 14.3(0.89)^t \)

The general form of an exponential model is \( P(t)=a(b)^t \), where \( a \) is the initial amount, \( b \) is the growth/decay factor. If \( 0 < b < 1 \), it's a decay model, and the decay rate \( r \) is given by \( 1 - b \).

Step2: Evaluate statement a and b

For the given model, \( b = 0.89 \). The decay rate \( r=1 - 0.89 = 0.11 \) or \( 11\% \). So the population is decreasing by \( 11\% \) each year. So statement a is false and statement b is true.

Step3: Evaluate statement c and d

When \( t = 0 \) (year 1992), \( P(0)=14.3(0.89)^0=14.3\times1 = 14.3 \) (in thousands of people). Since \( P(t) \) is in thousands of people, \( 14.3 \) thousand people is \( 14.3\times1000 = 14300 \) people. So statement c is false (it says 89000) and statement d is true.

Answer:

b) The population has decreased by 11% each year.
d) The population was 14,300 people in 1992.