Chapter 7.4, Problem 3E

a.

To determine

To find : an expression for the number of bacteria after $t$ hours

Answer to Problem 3E

The number of bacteria after $t$ hours is $100e^{\ln \left(4.2\right)t}$

Explanation of Solution

Given information : The question is

A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $\text{420}$

Calculation : The number of bacteria after $t$ hours is,

  $\begin{array}{l}B\left(0\right)=100\\ B\left(1\right)=420\\ B\left(t\right)=100e^{kt}\\ B\left(1\right)=420=100e^{1k}\\ \frac{420}{100}=4.2=e^k\\ \ln \left(4.2\right)=\ln \left(e^k\right)\\ \ln \left(4.2\right)=k\\ B\left(t\right)=100e^{\ln \left(4.2\right)t}\end{array}$

b.

To determine

To find : the number of bacteria after $3$ hours

Answer to Problem 3E

The number of bacteria after $3$ hours is $7408\text{ bacteria}$

Explanation of Solution

Given information The question is

A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $\text{420}$

Calculation :

  $\begin{array}{l}B\left(t\right)=100e^{\ln \left(4.2\right)t}\\ B\left(3\right)=100e^{\ln \left(4.2\right)\times 3}\\ B\left(3\right)=7408\text{ bacteria}\end{array}$

c.

To determine

To find : the rate growth after $3$ hours

Answer to Problem 3E

The rate growth after $3$ hours is $10,631$

Explanation of Solution

Given information : The question is

A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $\text{420}$

Calculation : the rate growth after $3$ hours is,

  $\begin{array}{l}\frac{dB}{dt}=kB\\ \frac{dB}{dt}=\ln \left(4.2\right)\times 7408\\ \frac{dB}{dt}=10,631\end{array}$

d.

To determine

When will the population reach $10,000$

Answer to Problem 3E

The population will reach $10,000$ in $3.2hours$

Explanation of Solution

Given information : The question is

A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $\text{420}$

Calculation :

  $\begin{array}{l}10,000=100e^{\ln \left(4.2\right)t}\\ \frac{10,000}{100}=e^{\ln \left(4.2\right)t}\\ \ln \left(100\right)=\ln \left(4.2\right)t\\ \frac{\ln \left(100\right)}{\ln \left(4.2\right)}=t\\ t=3.2hours\end{array}$