Answer :
To determine the correct function that represents the frog population after [tex]\( x \)[/tex] years when the population is decreasing at an average rate of [tex]\( 3\% \)[/tex] per year, we need to follow these steps:
1. Understand the Decrease in Population:
- The population is decreasing at a rate of [tex]\( 3\% \)[/tex] per year. This means that each year, the population retains [tex]\( 100\% - 3\% = 97\% \)[/tex] of its previous year's population.
2. Initial Population:
- When Ginny began her study, the frog population was estimated at 1,200.
3. Formulate the Recursive Model:
- Let the initial population be [tex]\( P_0 = 1200 \)[/tex].
- The population after one year would be [tex]\( P_1 = 1200 \times 0.97 \)[/tex].
- The population after two years would be [tex]\( P_2 = 1200 \times 0.97 \times 0.97 \)[/tex].
4. Generalize to [tex]\( x \)[/tex] Years:
- We can generalize this to say that the population after [tex]\( x \)[/tex] years, [tex]\( P(x) \)[/tex], is given by multiplying the initial population by [tex]\( 0.97 \)[/tex] raised to the power of [tex]\( x \)[/tex]:
[tex]\[ P(x) = 1200 \times (0.97)^x \][/tex]
Therefore, among the given choices, the function that correctly represents the frog population after [tex]\( x \)[/tex] years is:
[tex]\[ f(x) = 1,200 (0.97)^x \][/tex]
So the correct function choice is:
[tex]\[ \boxed{f(x) = 1,200 (0.97)^x} \][/tex]
This corresponds to the third option given in the question.
1. Understand the Decrease in Population:
- The population is decreasing at a rate of [tex]\( 3\% \)[/tex] per year. This means that each year, the population retains [tex]\( 100\% - 3\% = 97\% \)[/tex] of its previous year's population.
2. Initial Population:
- When Ginny began her study, the frog population was estimated at 1,200.
3. Formulate the Recursive Model:
- Let the initial population be [tex]\( P_0 = 1200 \)[/tex].
- The population after one year would be [tex]\( P_1 = 1200 \times 0.97 \)[/tex].
- The population after two years would be [tex]\( P_2 = 1200 \times 0.97 \times 0.97 \)[/tex].
4. Generalize to [tex]\( x \)[/tex] Years:
- We can generalize this to say that the population after [tex]\( x \)[/tex] years, [tex]\( P(x) \)[/tex], is given by multiplying the initial population by [tex]\( 0.97 \)[/tex] raised to the power of [tex]\( x \)[/tex]:
[tex]\[ P(x) = 1200 \times (0.97)^x \][/tex]
Therefore, among the given choices, the function that correctly represents the frog population after [tex]\( x \)[/tex] years is:
[tex]\[ f(x) = 1,200 (0.97)^x \][/tex]
So the correct function choice is:
[tex]\[ \boxed{f(x) = 1,200 (0.97)^x} \][/tex]
This corresponds to the third option given in the question.
