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:

Identify the initial value and decay rate

Using the Exponential Decay and Exponential Growth knowledge points

$$
LATEXBLOCK0
$$

Evaluate population estimates at specific times

Using the Exponential Decay knowledge point

$$
LATEXBLOCK1
$$

Analyze domain and range constraints

Using the Domain and Range knowledge point

$$
LATEXBLOCK2
$$

Answer:

• The function \(f(x) = 9,000(1.05)^x\) represents the situation.
• The function \(f(x) = 9,000(0.95)^x\) represents the situation. (Correct answer)
• After 2 years, the farmer can estimate that there will be about 8,120 bees remaining. (Correct answer)
• 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. (Correct answer)