Question

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?
f(x) = 1,200(0.97)^x
f(x) = 1,200(0.03)^x
f(x) = 1,200(0.97x)
f(x) = 1,200(1.03)^x

Explanation:

Step1: Recall exponential decay formula

The formula for exponential decay is \( f(x) = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decrease (in decimal), and \( x \) is time.

Step2: Identify values of \( a \) and \( r \)

Here, \( a = 1200 \) (initial population), \( r = 0.03 \) (3% decrease, so \( 1 - r = 1 - 0.03 = 0.97 \)).

Step3: Substitute into formula

Substituting \( a = 1200 \) and \( 1 - r = 0.97 \) into the decay formula gives \( f(x) = 1200(0.97)^x \).

Answer:

\( f(x) = 1,200(0.97)^x \)