- Expenential and Logarithmic Functions
Writing and evaluating a function modeling continuous exponential growt..
The number of bacteria in a culture decreases according to a continuous exponential decay model. The initial population in a study is 500 bacteria, and there are 290 bacteria left after 9 minutes.
(a) Let $t$ be the time (in minutes) since the beginning of the study, and let $y$ be the number of bacteria at time $t$.
Write a formula relating $y$ to $t$. Use exact expressions to fill in the missing parts of the formula. Do not use approximations.
$$y=◻e^{\mathbb{Q}t}$$
(b) How many bacteria are there 11 minutes after the beginning of the study?
Do not round any intermediate computations, and round your answer to the nearest whole number.
$◻$ bacteria
Explanation
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Step 1 ：Given the initial population $P=500$ bacteria and there are $290$ bacteria left after $9$ minutes, we can substitute these values into the formula $y=P{e}^{kt}$ to find the decay constant $k$.

Step 2 ：Substitute $P=500$, $y=290$, and $t=9$ into the formula, we get $290=500e^{9k}$.

Step 3 ：Solving for $k$, we get $e^{9k}=\frac{290}{500}=0.58$.

Step 4 ：Taking the natural logarithm of both sides, we get $9k=\ln (0.58)$.

Step 5 ：So, $k=\frac{\ln (0.58)}{9}$.

Step 6 ：Substituting $P=500$ and $k=\frac{\ln (0.58)}{9}$ back into the formula, we get the formula relating $y$ to $t$: $y=500e^{\frac{\ln (0.58)}{9}t}$.

Step 7 ：To find the number of bacteria 11 minutes after the beginning of the study, we substitute $t=11$ into the formula: $y=500e^{\frac{\ln (0.58)}{9}\cdot 11}$.

Step 8 ：Evaluating this expression, we get $y\approx 500e^{-0.511}\approx 224.5$.

Step 9 ：Rounding to the nearest whole number, we get $y\approx 225$.

Step 10 ：$\boxed{{\displaystyle 225}}$ is the approximate number of bacteria 11 minutes after the beginning of the study.