Question

a farmer estimates that he has 9,000 bees producing honey on his farm. the farmer becomes concerned when he realizes the population of bees seems to be decreasing steadily at a rate of 5% per year. if the number of bees in the population after x years is represented by f(x), which statements about the situation are true? check all that apply. the function f(x)=9,000(1.05)^x represents the situation. the function f(x)=9,000(0.95)^x represents the situation. after 2 years, the farmer can estimate that there will be about 8,120 bees remaining. after 4 years, the farmer can estimate that there will be about 1,800 bees remaining. the domain values, in the context of the situation, are limited to whole numbers. the range values, in the context of the situation, are limited to whole numbers.

Explanation:

Step1: Identify the decay - function formula

The general formula for exponential decay is $f(x)=a(1 - r)^x$, where $a$ is the initial amount, $r$ is the rate of decay, and $x$ is the number of time - periods. Here, $a = 9000$ and $r=0.05$. So the function is $f(x)=9000(1 - 0.05)^x=9000(0.95)^x$.

Step2: Calculate the number of bees after 2 years

Substitute $x = 2$ into $f(x)=9000(0.95)^x$. Then $f(2)=9000\times(0.95)^2=9000\times0.9025 = 8122.5\approx8120$.

Step3: Calculate the number of bees after 4 years

Substitute $x = 4$ into $f(x)=9000(0.95)^x$. Then $f(4)=9000\times(0.95)^4=9000\times0.81450625 = 7330.55625
eq1800$.

Step4: Analyze the domain and range

The number of years $x$ must be a non - negative whole number (you can't have a fraction of a year in this context for counting the number of years passed), so the domain values are limited to whole numbers. The number of bees must also be a whole number (you can't have a fraction of a bee), so the range values are limited to whole numbers.

Answer:

The function $f(x)=9000(0.95)^x$ represents the situation; After 2 years, the farmer can estimate that there will be about 8,120 bees remaining; The domain values, in the context of the situation, are limited to whole numbers; The range values, in the context of the situation, are limited to whole numbers.