Chapter 6.4, Problem 23E

(a)

To determine

To find: The number of bacteria will the colony contain at the end of $24hr$ .

Answer to Problem 23E

The answer is: $\approx 2.8\times {10}^{14}$ bacteria

Explanation of Solution

Given information:

Every half-hour, the colony grows by one bacterium and doubles in size.

Calculation:

The colony of bacteria adheres to the Law of Exponential Change proposed by

  $y=y_0{e}^{kt}$

Where,

  $\begin{array}{c}{y}_0=1,\\ t=24\text{h}\end{array}$

And $k$ is constant.

The concern mentions that the bacteria doubles every half hour. So, in $t=4$ there will be 4 bacteria.

Put these numbers in Equation $(1)$ and use the $\ln$ of both sides to solve for $k$ . Right now,

  $\begin{array}{c}4=(1)e^{k(1)}\\ \Rightarrow \ln 4=\ln e^k\\ \Rightarrow k=\ln 4\end{array}$

Put,

  $\begin{array}{c}k=4\\ t=24\end{array}$

To find $y$ for $24$ hours:

  $\begin{array}{c}y=(1)e^{(\ln 4)(24)}\\ \approx 2.8\times {10}^{14}\end{array}$

Therefore the required number of bacteria will the colony contain at the end of $24hr$ is

  $\approx 2.8\times {10}^{14}$ bacteria.

(b)

To determine

To explain: why, despite the fact that many bacteria are eliminated in an infected individual, a person who feels fine in the morning may become gravely ill by evening.

Answer to Problem 23E

Even if many of the bacteria are eliminated, there will still be enough since they reproduce quickly.

left to contaminate the person.

Explanation of Solution

Given information:

Every half-hour, the colony grows by one bacterium and doubles in size.

Explain:

The rate of growth of the bacterial population is exponential, which is substantially quicker than the rate of

Where by the body of the human can successfully handle incursions.

Even if many of the bacteria are eliminated, there will still be enough since they reproduce quickly.

Left to contaminate the person.