the number of wild flowers growing each year in a meadow is modeled by the function $f(x)$…

the number of wild flowers growing each year in a meadow is modeled by the function $f(x)$. $f(x)=\frac{1000}{1 + 9e^{-0.4x}}$. which statements are true about the population of wild flowers? select each correct answer. initially there were 100 wild flowers growing in the meadow. 42 more wildflowers will grow in the 11th year than in the 10th year. in the 15th year, there will be 1050 wild flowers in the meadow. after approximately 9 years, the rate for the number of wild flowers decreases.

the number of wild flowers growing each year in a meadow is modeled by the function $f(x)$. $f(x)=\frac{1000}{1 + 9e^{-0.4x}}$. which statements are true about the population of wild flowers? select each correct answer. initially there were 100 wild flowers growing in the meadow. 42 more wildflowers will grow in the 11th year than in the 10th year. in the 15th year, there will be 1050 wild flowers in the meadow. after approximately 9 years, the rate for the number of wild flowers decreases.

Answer

Explanation:

Step1: Find the initial number of wild - flowers

Set (x = 0) in (f(x)=\frac{1000}{1 + 9e^{-0.4x}}). Then (f(0)=\frac{1000}{1+9e^{0}}=\frac{1000}{1 + 9\times1}=\frac{1000}{10}=100). So the statement "Initially there were 100 wild flowers growing in the meadow" is true.

Step2: Calculate the number of wild - flowers in the 10th and 11th years

For (x = 10), (f(10)=\frac{1000}{1+9e^{-0.4\times10}}=\frac{1000}{1 + 9e^{-4}}\approx\frac{1000}{1+9\times0.0183}=\frac{1000}{1 + 0.1647}\approx859). For (x = 11), (f(11)=\frac{1000}{1+9e^{-0.4\times11}}=\frac{1000}{1+9e^{-4.4}}\approx\frac{1000}{1+9\times0.0123}=\frac{1000}{1+0.1107}\approx900). The difference (f(11)-f(10)\approx900 - 859 = 41\approx42). So the statement "42 more wildflowers will grow in the 11th year than in the 10th year" is true.

Step3: Analyze the number of wild - flowers in the 15th year

For (x = 15), (f(15)=\frac{1000}{1+9e^{-0.4\times15}}=\frac{1000}{1+9e^{-6}}\approx\frac{1000}{1+9\times0.0025}=\frac{1000}{1+0.0225}\approx978\neq1050). So the statement "In the 15th year, there will be 1050 wild flowers in the meadow" is false.

Step4: Analyze the rate of change

The function (y = f(x)=\frac{1000}{1+9e^{-0.4x}}) is a logistic - growth function. The derivative of (y) with respect to (x) is (y'=\frac{1000\times9\times0.4e^{-0.4x}}{(1 + 9e^{-0.4x})^2}). As (x) increases, the rate of change (y') first increases and then decreases. We can find the inflection - point of the logistic function. For a logistic function (y=\frac{L}{1+Ae^{-kx}}) (here (L = 1000), (A = 9), (k = 0.4)), the inflection - point occurs at (x=\frac{\ln(A)}{k}). Substituting (A = 9) and (k = 0.4), we get (x=\frac{\ln(9)}{0.4}\approx\frac{2.197}{0.4}\approx5.5). After the inflection - point, the rate of growth decreases. So the statement "After approximately 9 years, the rate for the number of wild flowers decreases" is true.

Answer:

Initially there were 100 wild flowers growing in the meadow. 42 more wildflowers will grow in the 11th year than in the 10th year. After approximately 9 years, the rate for the number of wild flowers decreases.