Question

writing functions for real - world problems
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: Determine the function formula

The initial number of bees is 9000 and it decreases at a rate of 5% (or 0.05) per year. 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 years. Here, $a = 9000$ and $r=0.05$, so $f(x)=9000(1 - 0.05)^x=9000(0.95)^x$. So the first statement is false and the second statement is true.

Step2: Calculate the number of bees after 2 years

Substitute $x = 2$ into $f(x)=9000(0.95)^x$. Then $f(2)=9000\times(0.95)^2=9000\times0.9025 = 8122.5\approx8120$. So the third statement is true.

Step3: Calculate the number of bees after 4 years

Substitute $x = 4$ into $f(x)=9000(0.95)^x$. Then $f(4)=9000\times(0.95)^4=9000\times0.81450625=7330.55625\approx7331
eq1800$. So the fourth statement is false.

Step4: Analyze the domain

The number of years $x$ can be 0, 1, 2, 3,... (non - negative whole numbers) in this context. So the domain values are limited to whole numbers, and the fifth statement is true.

Step5: Analyze the range

The number of bees must be a non - negative whole number (you can't have a fraction of a bee in a real - world count). So the range values are limited to whole numbers, and the sixth statement is true.

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.