rac{n}{5} > 2.095 \Rightarrow n > 10.475 \Rightarrow n \geq 11

February 22, 2026

$rac{n}{5} > 2.095 \Rightarrow n > 10.475 \Rightarrow n \geq 11$

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The annual growth rate is \( r = 3^{1/5} \). Now find the smallest integer \( n \) such that \( A_n > 10P_0 \):
r^n > 10 \Rightarrow (3^{1/5})^n > 10 \Rightarrow 3^{n/5} > 10 \Rightarrow rac{n}{5} > \log_3 10
Using \( \log_3 10 = rac{\ln 10}{\ln 3} pprox rac{2.3026}{1.0986} pprox 2.095 \), we get:
Thus, the smallest such \( n \) is 11.
oxed{r = 3^{1/5}, \; n = 11}
Question:** A high-altitude atmospheric carbon isotope tracking researcher uses quantum spectroscopy data modeled by the function \( f(x) = rac{x^3 - 3x}{x^2 + 1} \). Find the number of real solutions to the equation \( f(f(x)) = x \).

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