Question

a farmer estimates that he has 9,000 bees producing honey on his farm. the farmer determines conclusively that 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.

( square ) the function ( f(x)=9,000(1.05)^{x} ) represents the situation.

( square ) the function ( f(x)=9,000(0.95)^{x} ) represents the situation.

( square ) after 2 years, the farmer can estimate that there will be about 8,120 bees remaining.

( square ) after 4 years, the farmer can estimate that there will be about 1,800 bees remaining.

( square ) the domain values, in the context of the situation, are limited to whole numbers.

( square ) the range values, in the context of the situation, are limited to whole numbers.

Explanation:

Step1: Determine the function

The general formula for exponential decay is \(y = a(1 - r)^x\), where \(a\) is the initial amount, \(r\) is the rate of decay. Here, \(a = 9000\), \(r=0.05\). So \(f(x)=9000(1 - 0.05)^x=9000(0.95)^x\). The first option \(f(x)=9000(1.05)^x\) is for growth (\(r = 0.05\) growth), so it is wrong. The second option \(f(x)=9000(0.95)^x\) is correct.

Step2: Calculate for \(x = 2\)

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

Step3: Calculate for \(x = 4\)

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

Step4: Analyze domain and range

The domain \(x\) (years) is limited to whole - number values (you can't have a fraction of a year in this context of counting years passed). The range \(y = f(x)\) (number of bees) is also limited to whole numbers (you can't have a fraction of a bee in a real - world population count).

Answer:

The function \(f(x)=9000(0.95)^x\) represents the situation. After 2 years, the farmer can estimate that there will be about 8120 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. So the correct statements are:

• The function \(f(x)=9000(0.95)^x\) represents the situation.
• After 2 years, the farmer can estimate that there will be about 8120 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.