Questions: Use the approximate half-life formula for the case described below. Discuss whether the formula is valid for the case described. The production of a gold mine decreases by 5% per year. What is the approximate half-life for the production decline? Based on this approximate half-life and assuming that the mine's current annual production is 5000 kilograms, about what will its production be in 9 years? The half-life for the production decline is 14 years. (Type an integer or a decimal.) The production will be kilogram(s) in 9 years. (Round to the nearest whole number as needed.)

To solve this problem, we need to use the concept of exponential decay. The half-life formula can be used to determine the time it takes for the production to reduce to half its initial value. Given the annual decrease rate, we can calculate the production after a certain number of years using the exponential decay formula.

1. Calculate the decay constant: Use the given half-life to find the decay constant.
2. Calculate the production after 9 years: Use the decay constant to find the production after 9 years.

Die anfängliche Produktion beträgt 5000 Kilogramm. Die Halbwertszeit beträgt 14 Jahre. Wir berechnen die Zerfallskonstante \( k \) mit der Formel: \[ k = \frac{\ln(2)}{\text{Halbwertszeit}} \] \[ k = \frac{\ln(2)}{14} \approx 0{,}04951 \]

Wir verwenden die Zerfallskonstante \( k \) und die exponentielle Zerfallsformel, um die Produktion nach 9 Jahren zu berechnen: \[ P(t) = P_0 \cdot e^{-kt} \] \[ P(9) = 5000 \cdot e^{-0{,}04951 \cdot 9} \] \[ P(9) \approx 3202 \]

Die Produktion nach 9 Jahren beträgt \(\boxed{3202}\) Kilogramm.