Question

A culture of the bacterium Salmonella enteritidis initially contains $50$ cells.

When introduced into a nutrient broth, the culture grows at a rate proportional to its size.

After $1.5$ hours, the population has increased to $775$.

$\left(a\right)$ Find an expression for the number of bacteria after $t$ hours. (Round your numeric values to four decimal places.) $P\left(t\right)=$ ________

$\left(b\right)$ Find the number of bacteria after $5$ hours. (Round your answer to the nearest whole number.) $P\left(5\right)=$______ bacteria

$\left(c\right)$ Find the rate of growth (in bacteria per hour) after $5$ hours. (Round your answer to the nearest whole number.)$P\text{'}\left(5\right)=$ ______ bacteria per hour

$\left(d\right)$ After how many hours will the population reach $250,000$? (Round your answer to one decimal place.) $t=$ _______ hr.

Answer 1

Step-1: Find an expression for the number of bacteria after $t$ hours:

The exponential growth model is $P=P_o{e}^{rt}$.

Where,

$P_o=$Initial growth

$r=$growth rate

$t=$time period

It is given that initially cells, $P_o=50$

time period, $t=1.5$ hours

Population after $t=1.5$ hours is, $P=775$.

Substitute the given values in the growth exponential model:

$$\begin{gathered} 775=50e^{r\times 1.5} \\ \Rightarrow \frac{775}{50}=e^{1.5r} \\ \Rightarrow 15.5=e^{1.5r} \\ \Rightarrow \log \left(15.5\right)=\log e^{1.5r} \\ \Rightarrow \log \left(15.5\right)=1.5r\left[\because \log e^x=x\right] \\ \Rightarrow 1.19=1.5r \\ \Rightarrow \frac{1.19}{1.5}=r \\ \therefore r=0.8\% \end{gathered}$$

Thus, $P\left(t\right)=50e^{0.8t}$

Step-2: Find the number of bacteria after $5$ hours.

Number of bacteria after $5$ hours is calculated as follow:

$$\begin{gathered} P\left(5\right)=50e^{0.8\times 5} \\ =2729.9 \end{gathered}$$

Step-3: Find the rate of growth (in bacteria per hour) after $5$ hours.

$P\left(t\right)=50e^{0.8t}$

Differentiate w.r.t $t$:

$P\text{'}\left(t\right)=50\times 0.8e^{0.8t}$

Rate of growth (in bacteria per hour) after $5$ hours is as follow:

$$\begin{gathered} P\text{'}\left(5\right)=50\times 0.8e^{0.8\times 5} \\ =440.9 \end{gathered}$$

Step-4: Find time $t$.

The population at time $t$ is $P\left(t\right)=250,000$

So,

$$\begin{gathered} 250,000=50e^{0.8t} \\ \Rightarrow \frac{250,000}{50}=e^{0.8t} \\ \Rightarrow 5000=e^{0.8t} \\ \Rightarrow 0.8t=3.7 \\ \Rightarrow t=\frac{3.7}{0.8} \\ \therefore t=4.6 \end{gathered}$$

Hence, The required expression for blanks are shown below,

(a) $P\left(t\right)=50e^{0.8t}$

(b) $P\left(5\right)=2729.9$

(c) $P\text{'}\left(5\right)=440.9$

(d) $t=4.6$

A culture of the bacterium Salmonella enteritidis initially contains $50$ cells.

When introduced into a nutrient broth, the culture grows at a rate proportional to its size. After $1.5$ hours, the population has increased to $775$.

(a) Find an expression for the number of bacteria after $t$ hours. (Round your numeric values to four decimal places.) $\underline{P\left(t\right)=50e^{0.8t}}$

(b) Find the number of bacteria after $5$ hours. (Round your answer to the nearest whole number.) $\underline{P\left(5\right)=2729.9}$ bacteria

(c) Find the rate of growth (in bacteria per hour) after $5$ hours. (Round your answer to the nearest whole number.)$\underline{P\text{'}\left(5\right)=440.9}$ bacteria per hour

(d) After how many hours will the population reach $250,000$? (Round your answer to one decimal place.) $\underline{\mathrm{t}=4.6\mathrm{hr}.}$