Question

mathematics i a-g cr sem a aguirre
exponential decay functions
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? choose three correct answers.
after 4 years, the farmer
can estimate that there will
be about 1,800 bees
remaining.
the function
f(x) = 9,000(1.05)^x
represents the situation.
the range values, in the
context of the situation, are
limited to whole numbers.
the domain values, in the
context of the situation, are
limited to whole numbers.
after 2 years, the farmer
can estimate that there will
be about 8,120 bees
remaining.
the function
f(x) = 9,000(0.95)^x
represents the situation.

Explanation:

Step1: Analyze the exponential decay model

The general form of an exponential decay function is \( f(x)=a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decay, and \( x \) is the time. Here, \( a = 9000 \) and \( r=0.05 \), so the function should be \( f(x)=9000(0.95)^x \). So the sixth statement is correct, and the second is wrong.

Step2: Check the number of bees after 2 years

Substitute \( x = 2 \) into \( f(x)=9000(0.95)^x \). Calculate \( (0.95)^2=0.9025 \), then \( f(2)=9000\times0.9025 = 8122.5\approx8120 \). So the fifth statement is correct.

Step3: Check the number of bees after 4 years

Substitute \( x = 4 \) into \( f(x)=9000(0.95)^x \). Calculate \( (0.95)^4=(0.95^2)^2 = 0.9025^2=0.81450625 \), then \( f(4)=9000\times0.81450625 = 7330.55625\approx7331 \), not 1800. So the first statement is wrong.

Step4: Analyze domain and range

• Domain: In the context of years, \( x \) can be non - negative real numbers (e.g., partial years), but it is often considered as non - negative integers in practical terms (since we count years as whole numbers). However, strictly speaking, the domain of the function \( f(x)=9000(0.95)^x \) is all real numbers \( x\geq0 \), but in the context of the situation (number of years), it can be considered as non - negative integers (whole numbers). So the fourth statement: "The domain values, in the context of the situation, are limited to whole numbers" is correct.
• Range: The number of bees \( f(x) \) represents a count, so it should be a whole number (you can't have a fraction of a bee). So the third statement: "The range values, in the context of the situation, are limited to whole numbers" is correct.

Answer:

The three correct statements are:

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

(We choose three from the correct ones: "The range values, in the context of the situation, are limited to whole numbers", "The domain values, in the context of the situation, are limited to whole numbers", "After 2 years, the farmer can estimate that there will be about 8,120 bees remaining", "The function \( f(x)=9,000(0.95)^x \) represents the situation" (any three of these four correct ones))