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? choose three correct answers.
the function $f(x)=9,000(1.05)^x$ represents the situation.
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 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
the range values, in the context of the situation, are limited to whole numbers

Explanation:

Step1: Identify decay function form

For a 5% annual decrease, the remaining percentage each year is $1 - 0.05 = 0.95$. The exponential decay function is $f(x) = 9000(0.95)^x$.

Step2: Verify domain/range constraints

Domain (years $x$) can be non-negative real numbers (we can calculate for partial years), but range (number of bees) must be whole numbers (you can't have a fraction of a bee).

Step3: Calculate bees after 4 years

Substitute $x=4$ into $f(x)$:
$f(4) = 9000(0.95)^4 = 9000 \times 0.81450625 = 7330.55625 \approx 7331$

Step4: Calculate bees after 2 years

Substitute $x=2$ into $f(x)$:
$f(2) = 9000(0.95)^2 = 9000 \times 0.9025 = 8122.5 \approx 8123$, which is approximately 8120.

Step5: Evaluate all statements

• $f(x)=9000(1.05)^x$ is growth, false.
• 4-year estimate ~1800 is false.
• Domain limited to whole numbers is false.
• $f(x)=9000(0.95)^x$ is true.
• 2-year estimate ~8120 is true.
• Range limited to whole numbers is true.

Answer:

1. The function $f(x) = 9,000(0.95)^x$ represents the situation
2. After 2 years, the farmer can estimate that there will be about 8,120 bees remaining
3. The range values, in the context of the situation, are limited to whole numbers