Pregunta
A certain type of bacteria decreases according to the model \( P(t)=12,789 \mathrm{e}^{-0.45 t} \), where \( P \) is the population of the bacteria after \( t \) minutes. What will be the eventual population of this sample as time \( t \) approaches infinity (round to the nearest whole bacteria)? 685 5,865 1,865 0
Ask by Reeves Deleon. in the United States
Jan 25,2025
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As time increases, the bacteria population approaches 0.
**Answer:** 0
Solución
To determine the eventual population of the bacteria as time \( t \) approaches infinity, we'll analyze the given exponential decay model:
\[
P(t) = 12,789 \, \mathrm{e}^{-0.45 t}
\]
As \( t \) becomes very large (\( t \to \infty \)), the exponential term \( \mathrm{e}^{-0.45 t} \) behaves as follows:
\[
\lim_{t \to \infty} \mathrm{e}^{-0.45 t} = 0
\]
Therefore, the population \( P(t) \) approaches:
\[
\lim_{t \to \infty} P(t) = 12,789 \times 0 = 0
\]
**Answer:** 0
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The Deep Dive
As time \( t \) approaches infinity, the term \( e^{-0.45t} \) approaches zero. This means that the population \( P(t) \) will decrease and ultimately get closer and closer to zero. Therefore, the eventual population of this sample will be approximately 0 bacteria. So, the answer is 0.
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