Chapter 4.6, Problem 16E

a.

To determine

Find the population grows exponentially.

Answer to Problem 16E

  $\boxed{2029}$

Explanation of Solution

Given information:

The population of the world was $5.7$ billion in $1995$ , and the observed relative growth rate was $2\%$ per year.

By what year will the population have doubled?

Calculation:

Consider the population growth model,

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ where,\\ n_0=initial-population\\ r=growth-rate\\ t=time\end{array}$

The population of the world was $5.7$ billion in $1995$ ,

  $\begin{array}{l}5.7\times 2=5.7e^{0.02t}\\ 11.4=5.7e^{0.02t}\\ e^{0.02t}=2\\ \ln e^{0.02t}=\ln 2\\ t=\ln 2/0.02\\ =34.65\end{array}$

The population have doubled in,

  $1995+34=2029$

Hence, the solution is, $\boxed{2029}$

b.

To determine

Find the population grows exponentially.

Answer to Problem 16E

  $\boxed{2049}$

Explanation of Solution

Given information:

The population of the world was $5.7$ billion in $1995$ , and the observed relative growth rate was $2\%$ per year.

By what year will the population have tripled?

Calculation:

Consider the population growth model,

  $\begin{array}{l}n\left(t\right)=n_0{e}^{rt}\\ where,\\ n_0=initial-population\\ r=growth-rate\\ t=time\end{array}$

The population of the world was $5.7$ billion in $1995$ ,

  $\begin{array}{l}5.7\times 3=5.7e^{0.02t}\\ 17.1=5.7e^{0.02t}\\ e^{0.02t}=3\\ \ln e^{0.02t}=\ln 3\\ t=\ln 3/0.02\\ =54.93\approx 55\end{array}$

The population have tripled in,

  $1995+54=2049$

Hence, the solution is, $\boxed{2049}$