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.

Answer 1

The function f(x)=9,000(0.95)x represents the situation.

After years, the farmer can estimate that there will be about 8,120 bees remaining.

The range value, in the context of the situation, are limited to whole numbers.

Answer 2

Analysis to obtain the function that models the polulaiton ob bees:

1) First year 9,000 bees

2) Second year: decrease 5% => 9,000 - 0.05* 9,000 = 9,000 * (1 - 0.05) = 9,000 * 0.95

3) Every year the population decreases 5% => 9,000 * 0.95)^ (number of years)

4) if you call x the number of years, and f(x) the function that represents the number of bees, then: f(x) = 9,000 (0.95)^ x.

Analysis of the statements:

1) The function f(x) = 9,000(1.05)x represents the situation.

FALSE: WE DETERMINED IT IS f(x) = 9,000 (0.95)^x

2) The function f(x) = 9,000(0.95)x represents the situation.

TRUE: THAT IS WHAT WE OBTAINED AS CONCLUSION OF THE PREVIOUS ANALYSIS.

3) After 2 years, the farmer can estimate that there will be about 8,120 bees remaining.

Do the math:

f(2) = 9,000 * (0.95)^2 = 9,000 * 0,9025 = 8,122

So, the statement is TRUE

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

f(4) = 9,000 * (0.95)^4 = 9,000 * 0.81450625 = 7,330

So, the statement is FALSE

5) The domain values, in the context of the situation, are limited to whole numbers.

FALSE: THE DOMAIN VALUES ARE ALL NON NEGATIVE REAL VALUES. FOR EXAMPLE THE FUNCTION IS WELL DEFINED FOR X = 5 AND HALF

6) The range values, in the context of the situation, are limited to whole numbers.

TRUE: THERE CANNOT BE FRACTIONS OF BEES