Question

the exponential model below represents the population of an animal ( x ) years from now. what is meaning of 9000 in the model?
f(x) = 9000(1.02)^x
show your work here

• every year, the animal increases in population by 9000
• the animals population will be zero after 9000 years
• the animals population today is 9000
• every year, the animal decreases in population by 9000

Explanation:

Step1: Recall exponential model form

The general exponential growth model is \( f(x) = a(b)^x \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( x \) is time.

Step2: Analyze the given function

In \( f(x) = 9000(1.02)^x \), \( x \) represents years from now. When \( x = 0 \) (today, since 0 years from now), \( f(0)=9000(1.02)^0 = 9000(1)=9000 \). So 9000 is the initial population (today's population).

Step3: Evaluate options

• "Every year, the animal increases in population by 9000": The growth is by a factor, not a constant 9000. Eliminate.
• "The animal’s population will be zero after 9000 years": The model shows growth, not decline to zero. Eliminate.
• "The animal’s population today is 9000": Matches our analysis (x=0 gives 9000).
• "Every year, the animal decreases in population by 9000": The base \( 1.02>1 \), so it's growth, not decrease. Eliminate.

Answer:

The animal’s population today is 9000 (the corresponding option: "The animal’s population today is 9000")