The number of bacteria in a culture is increasing according to the law of exponential growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 bacteria in the culture. Answer the following questions, rounding numerical answers to four decimal places.
(i) Find the initial population.
(ii) Write an exponential growth model for the bacteria population. Let $t$ represent time in hours.
(iii) Use the model to determine the number of bacteria after 20 hours.
(iv) After how many hours will the bacteria count be 15,000 ?
A) (i) $72.0588$; (ii) $y = 72.0588 e ^ { 0.1987 t }$; (iii) $3,819.3668$; (iv) $32.4162 \mathrm { hr }$
B) (i) $74.2088$; (ii) $y = 74.2088 e ^ { 0.2338 t }$; (iii) $5,194.0840$; (iv) $34.6442 \mathrm { hr }$
C) (i) $72.0588$; (ii) $y = 72.0588 e ^ { 0.1775 t }$; (iii) $2,506.6327$; (iv) $30.0817 \mathrm { hr }$
D) (i) $77.8388$; (ii) $y = 77.8388 e ^ { 0.2409 t }$;(iii) $7,945.5374$;(iv) $36.7554 \mathrm { hr }$
E) (i) $79.3988$; (ii) $y = 79.3988 e ^ { 0.2598 t }$; (iii) $10.598 .0009$;(iv) $38.5348 \mathrm { hr }$

Question

Quizplus · Accepted Answer

(i) Using the formula $y = y_0 e^{kt}$ and the information given, we can set up a system of equations:
\begin{align*}
175 &= y_0 e^{5k} \
425 &= y_0 e^{10k}
\end{align*}
Dividing the second equation by the first gives:
\begin{align*}
\frac{425}{175} &= \frac{y_0 e^{10k}}{y_0 e^{5k}} \
2.4286 &= e^{5k} \
k &= \frac{\ln 2.4286}{5} \approx 0.1775
\end{align*}
Substituting this value of $k$ into the first equation above gives:
\begin{align*}
175 &= y_0 e^{5(0.1775)} \
y_0 &\approx 72.0588
\end{align*}
Therefore, the initial population is approximately 72.0588.
(ii) The exponential growth model is:
$$y = 72.0588 e^{0.1775t}$$
(iii) Plugging in $t=20$ gives:
$$y = 72.0588 e^{0.1775(20)} \approx 2,506.6327$$
Therefore, the number of bacteria after 20 hours is approximately 2,506.6327.
(iv) We solve for $t$ in the equation $15,000 = 72.0588 e^{0.1775t}$:
\begin{align*}
\frac{15,000}{72.0588} &= e^{0.1775t} \
\ln \frac{15,000}{72.0588} &= 0.1775t \
t &\approx 30.0817
\end{align*}
Therefore, after approximately 30.0817 hours, the count will be 15,000.

The Number of Bacteria in a Culture Is Increasing According ttt

The Number of Bacteria in a Culture Is Increasing According $t$