Question

question 11 · 1 point
a sample of bacteria is decaying according to an exponential decay model. if the sample begins with 600 bacteria, and after 15 minutes there are 360 bacteria, after how many minutes will there be 20 bacteria remaining?
when solving this problem, round the value of (k) to four decimal places and round your final answer to the nearest whole number.
provide your answer below:

minutes

Explanation:

Step1: Write the exponential - decay formula

$A = A_0e^{-kt}$, where $A$ is the final amount, $A_0$ is the initial amount, $k$ is the decay constant, and $t$ is the time.
We know that $A_0 = 600$, $t = 15$ minutes, and $A = 360$. Substitute these values into the formula:
$360=600e^{-15k}$

Step2: Solve for $k$

First, divide both sides of the equation by 600:
$\frac{360}{600}=e^{-15k}$
$0.6 = e^{-15k}$
Take the natural - logarithm of both sides:
$\ln(0.6)=\ln(e^{-15k})$
Since $\ln(e^x)=x$, we have $\ln(0.6)=-15k$.
$k=-\frac{\ln(0.6)}{15}\approx-\frac{- 0.5108}{15}\approx0.0341$

Step3: Find the time $t$ when $A = 20$

Substitute $A_0 = 600$, $A = 20$, and $k\approx0.0341$ into the exponential - decay formula $A = A_0e^{-kt}$:
$20 = 600e^{-0.0341t}$
Divide both sides by 600:
$\frac{20}{600}=e^{-0.0341t}$
$\frac{1}{30}=e^{-0.0341t}$
Take the natural - logarithm of both sides:
$\ln(\frac{1}{30})=\ln(e^{-0.0341t})$
Since $\ln(e^x)=x$, we have $\ln(\frac{1}{30})=-0.0341t$
$t =-\frac{\ln(\frac{1}{30})}{0.0341}=\frac{\ln(30)}{0.0341}\approx\frac{3.4012}{0.0341}\approx112$

Answer:

112