Question
Question A sample of bacteria is decaying according to the half-life model \( f(t)=N e^{-0.220 t} \), where \( N \) is the initic is time in hours. At \( t=4 \), there are 25 bacteria remaining. What was the initial sample amount? Round your final answer to the nearest whole number. Do not include units.
Ask by Parsons Mathis. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The initial sample amount was approximately 60 bacteria.
Solution
We start with the decay model
\[
f(t) = N e^{-0.220t}
\]
Given that at \( t=4 \) there are 25 bacteria, we substitute to obtain
\[
25 = N e^{-0.220 \cdot 4}.
\]
Simplify the exponent:
\[
-0.220 \times 4 = -0.88,
\]
so that
\[
25 = N e^{-0.88}.
\]
To solve for \( N \), divide both sides by \( e^{-0.88} \):
\[
N = \frac{25}{e^{-0.88}} = 25 e^{0.88}.
\]
Now, calculate \( e^{0.88} \). Using a calculator,
\[
e^{0.88} \approx 2.410.
\]
Thus,
\[
N \approx 25 \times 2.410 \approx 60.25.
\]
Rounding to the nearest whole number gives
\[
N \approx 60.
\]
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Simplify this solution Bonus Knowledge
To determine the initial sample amount of bacteria, we can use the provided formula. At \( t=4 \), we know the remaining number of bacteria is 25. We plug this into the equation and solve for \( N \): \[ f(t) = N e^{-0.220 \cdot 4} = 25 \] Calculating \( e^{-0.220 \cdot 4} \): \[ e^{-0.880} \approx 0.414 \] Now substituting this value back into the equation gives us: \[ N \cdot 0.414 = 25 \] Now, solving for \( N \): \[ N = \frac{25}{0.414} \approx 60.4 \] Rounding to the nearest whole number, the initial sample amount was approximately **60**.
