Chapter 7.4, Problem 23E

a.

To determine

To Find: The numbers of bacteria will the colony contain at the end of $\text{24h}$

Answer to Problem 23E

The numbers of bacteria will the colony contain at the end of $\text{24h}$

  $\approx 2.815\times {10}^{14}$

Explanation of Solution

Given information: The cholera bacteria in a colony grows unchecked according to the law of exponential change. The colony starts with $\text{1}$ bacterium and doubles in number every half hour

Formula used:

  $y=y_0{e}^{-kt}$

Where $k$ is a constant

Calculation:The initial condition is $y_0=1$ bacteria

Bacterium doubles every half hour

So, in $\text{t=}\frac{\text{1}}{\text{2}}$ hour there are $\text{2}$ bacteria

  $\begin{array}{l}y=y_0{e}^{-kt}\\ 2=e^{\frac{-k}{2}}\\ k=-\ln 4\end{array}$

The numbers of bacteria at $\text{24h}$

  $\begin{array}{l}y=y_0{e}^{-kt}\\ y=e^{24\ln 4}\\ y\approx 2.815\times {10}^{14}\end{array}$

b.

To determine

To Explain: Why a person who feels well in the morning may be dangerously ill by evening even though, in an infected person

Answer to Problem 23E

The overall number of bacteria still increase in time, and thus a well person may be dangerously ill by evening

Explanation of Solution

Given information: The numbers of bacteria will the colony contain at the end of $\text{24h}$

  $\approx 2.815\times {10}^{14}$

Bacteria populations grows at an exponential rate

  $y=y_0{e}^{-kt}$

The bacteria reproduce fast enough that even if many are destroyed there are still enough left to make the person sick. The overall number of bacteria still increase in time, and thus a well person may be dangerously ill by evening.