Chapter 1.7, Problem 69E

(a)

To determine

To find the doubling time of $S_1\left(t\right)$ using the given values

Answer to Problem 69E

Therefore, the doubling time is $3.5unit$ .

Explanation of Solution

Given information:

$S_1\left(t\right)=3.4e^{0.2t}$.

Calculation:

We are to find the doubling time and plot graph to check.

This measurement starts at $S\left(0\right)=3.4$ and doubles when $S\left(t\right)=6.8$

Therefore,

$\begin{array}{l}3.4e^{0.2t}=6.8\\ e^{0.2t}=2\\ In{e}^{0.2t}=In2\\ 0.2t=0.6931\\ t=\frac{0.6931}{0.2}\\ =3.5\end{array}$

Conversely,

$S\left(3.5\right)=3.4e^{0.2\left(3.5\right)}=6.8$

Therefore, the doubling time is $3.5unit$

And the graph showing the doubling time is drawn below:

  

(b)

To determine

To find the doubling time of $S_2\left(t\right)$ using the given values

Answer to Problem 69E

Therefore, the doubling time is $0.2unit$ .

Explanation of Solution

Given information:

$S_2\left(t\right)=0.2e^{3.4t}$.

Calculation:

We are to find the doubling time and plot graph to check.

This measurement starts at $S\left(0\right)=.2$ and doubles when $S\left(t\right)=.4$

Therefore,

$\begin{array}{l}0.2e^{3.4t}=.4\\ e^{3.4t}=2\\ In{e}^{3.4t}=In2\\ 3.4t=0.6931\\ t=\frac{0.6931}{3.4}\\ =0.2\end{array}$

Conversely,

$S\left(0.2\right)=0.2e^{0.2t}=.4$

Therefore, the doubling time is $0.2unit$

And the graph showing the doubling time is drawn below:

  

(c)

To determine

To find the half-life of $H_1\left(t\right)$ using the given values

Answer to Problem 69E

The half life is $3.5unit$

Explanation of Solution

Given information:

$H_1\left(t\right)=3.4e^{-0.2t}$.

Calculation:

We are to find the half life and plot graph to check.

To find the half life, we have

  $\begin{array}{l}3.4e^{-0.2t}=1.7\\ e^{-0.2t}=.5\\ In{e}^{-0.2t}=In.5\\ -0.2t=.6931\\ t=\frac{-0.6931}{-0.2}\\ =3.5\end{array}$

Conversely,

$S\left(3.5\right)=3.4e^{-0.2(3.5)}=1.7$

Therefore, the half life is $3.5unit$

And the graph showing the half life is drawn below:

  

(d)

To determine

To find the half-life of $H_2\left(t\right)$ using the given values

Answer to Problem 69E

$0.2unit$ .

Explanation of Solution

Given information: $H_2\left(t\right)=0.2e^{-3.4t}$.

Calculation:

We are to find the half life and plot graph to check.

To find the half life, we have

  $\begin{array}{l}0.2e^{-3.4t}=.1\\ e^{-3.4t}=.5\\ In{e}^{-3.4t}=In.5\\ -3.4t=-0.6931\\ t=\frac{-0.6931}{-3.4}\\ =0.2\end{array}$

Conversely,

$S\left(0.2\right)=0.2e^{0.2t}=0.1$

Therefore, the half life is $0.2unit$

And the graph showing the half life is drawn below: