Ginny is studying a population of frogs. She determines that the population is decreasing at an average rate of $3 \%$ per year. When she began her study, the frog population was estimated at 1,200. Which function represents the frog population after $x$ years?
A. [tex]f^{\prime}(x)=1,200(1.03)^x[/tex]
B. [tex]f(x)=1,200(0.03)^x[/tex]
C. [tex]f(x)=1,200(0.97)^x[/tex]
D. [tex]f(x)=1,200(0.97 x)[/tex]
Answer (2)
The problem describes an exponential decay situation where a frog population decreases by 3% annually.
The general formula for exponential decay is f ( x ) = a ( 1 − r ) x , where a is the initial amount and r is the decay rate.
Substituting the given values, the function becomes f ( x ) = 1200 ( 1 − 0.03 ) x .
Simplifying, the function representing the frog population after x years is f ( x ) = 1 , 200 ( 0.97 ) x .
Explanation
Understanding the Problem We are given that the initial frog population is 1,200 and it decreases at a rate of 3% per year. We need to find a function that represents the population after x years.
General Exponential Decay Function The general form of an exponential decay function is given by: f ( x ) = a ( 1 − r ) x where:
f ( x ) is the population after x years,
a is the initial population,
r is the rate of decay (as a decimal),
x is the number of years.
Identifying the Values In this problem, we have:
Initial population, a = 1200
Rate of decay, r = 3% = 0.03
Substituting the Values Substitute the given values into the exponential decay function: f ( x ) = 1200 ( 1 − 0.03 ) x f ( x ) = 1200 ( 0.97 ) x
Final Answer Therefore, the function that represents the frog population after x years is f ( x ) = 1 , 200 ( 0.97 ) x .
Examples
Exponential decay is a concept used to model various real-world phenomena, such as the decrease in the value of a car over time. Suppose you buy a new car for 25 , 000 , an d i t s v a l u e d e p rec ia t es a t a r a t eo f 10 x ye a rsc anb e m o d e l e d b y t h e f u n c t i o n V(x) = 25000(0.9)^x$. This helps you understand how much the car is worth after a certain period and when you might consider selling or trading it in. Similarly, this concept applies to population decreases, radioactive decay, and other areas where quantities decrease over time.
The frog population, initially at 1,200 and decreasing by 3% per year, is modeled by the function f ( x ) = 1200 ( 0.97 ) x . This function represents exponential decay, giving the population after x years. The correct answer is option C: f ( x ) = 1 , 200 ( 0.97 ) x .
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