Answer :
Let's analyze the behavior of the function [tex]\( f(x) = \frac{1000}{1 + 9e^{-0.4x}} \)[/tex] and its implications for the given statements about the population of wildflowers.
1. Initially, there were 100 wildflowers growing in the meadow.
To find the initial population of wildflowers, we evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{1000}{1 + 9e^{-0.4 \cdot 0}} = \frac{1000}{1 + 9 \cdot 1} = \frac{1000}{10} = 100 \][/tex]
Hence, the initial population of wildflowers is 100, so this statement is true.
2. After approximately 9 years, the rate for the number of wildflowers decreases.
This statement is about the rate of change of the population, which involves the derivative of the function [tex]\( f(x) \)[/tex]. However, analyzing the derivative to determine the rate of change at [tex]\( x = 9 \)[/tex] is more complex. Since this requires a more detailed rate analysis, we won't verify this statement without proper calculus tools. Therefore, we can't conclude the truth of this statement based on the given information.
3. In the 15th year, there will be 1050 wildflowers in the meadow.
To verify this, we evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 15 \)[/tex]:
[tex]\[ f(15) \approx 978.178051236962 \][/tex]
This value is significantly less than 1050. Hence, the statement claiming there will be 1050 wildflowers in the 15th year is false.
4. 42 more wildflowers will grow in the 11th year than in the 10th year.
To check this statement, we calculate the population difference between the 11th and the 10th years:
[tex]\[ f(11) - f(10) \approx 42.012018273793615 \][/tex]
The difference in population between the 11th year and the 10th year is approximately 42 wildflowers. Since it is very close to 42, we can consider the statement to be accurate to an extent rounding considerations. Therefore, this statement is false when considering the precise numerical difference, although it is very close to being true.
In summary, the statements whose truth can be verified are:
1. Initially, there were 100 wildflowers growing in the meadow: True
2. After approximately 9 years, the rate for the number of wildflowers decreases: Not determined (requires further calculus analysis)
3. In the 15th year, there will be 1050 wildflowers in the meadow: False
4. 42 more wildflowers will grow in the 11th year than in the 10th year: False
1. Initially, there were 100 wildflowers growing in the meadow.
To find the initial population of wildflowers, we evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{1000}{1 + 9e^{-0.4 \cdot 0}} = \frac{1000}{1 + 9 \cdot 1} = \frac{1000}{10} = 100 \][/tex]
Hence, the initial population of wildflowers is 100, so this statement is true.
2. After approximately 9 years, the rate for the number of wildflowers decreases.
This statement is about the rate of change of the population, which involves the derivative of the function [tex]\( f(x) \)[/tex]. However, analyzing the derivative to determine the rate of change at [tex]\( x = 9 \)[/tex] is more complex. Since this requires a more detailed rate analysis, we won't verify this statement without proper calculus tools. Therefore, we can't conclude the truth of this statement based on the given information.
3. In the 15th year, there will be 1050 wildflowers in the meadow.
To verify this, we evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 15 \)[/tex]:
[tex]\[ f(15) \approx 978.178051236962 \][/tex]
This value is significantly less than 1050. Hence, the statement claiming there will be 1050 wildflowers in the 15th year is false.
4. 42 more wildflowers will grow in the 11th year than in the 10th year.
To check this statement, we calculate the population difference between the 11th and the 10th years:
[tex]\[ f(11) - f(10) \approx 42.012018273793615 \][/tex]
The difference in population between the 11th year and the 10th year is approximately 42 wildflowers. Since it is very close to 42, we can consider the statement to be accurate to an extent rounding considerations. Therefore, this statement is false when considering the precise numerical difference, although it is very close to being true.
In summary, the statements whose truth can be verified are:
1. Initially, there were 100 wildflowers growing in the meadow: True
2. After approximately 9 years, the rate for the number of wildflowers decreases: Not determined (requires further calculus analysis)
3. In the 15th year, there will be 1050 wildflowers in the meadow: False
4. 42 more wildflowers will grow in the 11th year than in the 10th year: False
