The function $f(x)=1200(0.8)^x$ represents the possible elk population of a national park $x$ years from now. What is the current population of elk at the park?
A. 960
B. 768
C. 1500
D. 1200
Answer (1)
Substitute x = 0 into the function f ( x ) = 1200 ( 0.8 ) x .
Calculate ( 0.8 ) 0 = 1 .
Multiply 1200 by 1 to find the current population.
The current population of elk is 1200 .
Explanation
Understanding the Problem The problem states that the function f ( x ) = 1200 ( 0.8 ) x models the elk population of a national park x years from now. We need to find the current population, which means we need to find the population at x = 0 (now).
Substituting x=0 To find the current population, we substitute x = 0 into the function: f ( 0 ) = 1200 ( 0.8 ) 0
Calculating the Population Since any number (except 0) raised to the power of 0 is 1, we have ( 0.8 ) 0 = 1 . Therefore, f ( 0 ) = 1200 × 1 = 1200
Final Answer The current population of elk at the park is 1200.
Examples
Understanding exponential functions like the one in this problem can help predict population growth or decay in various scenarios, such as bacterial cultures, wildlife populations, or even financial investments. For example, if a bacteria culture starts with 1000 bacteria and doubles every hour, the population after t hours can be modeled by P ( t ) = 1000 × 2 t . Similarly, in finance, compound interest can be modeled using an exponential function to predict the growth of an investment over time. These models help in making informed decisions in ecology, medicine, and finance.
