Chapter 3.1, Problem 68E

a.

To determine

To find if the population of Italy increasing or decreasing?

Answer to Problem 68E

  $\boxed{Increasing}$

Explanation of Solution

Given information:

The populations (in millions) of Italy from $2000$ through $2012$ can be approximated by the model $p=57.563e^{0.0052t}$ , where $t$ represents the year, with $t=0$ corresponding to $2000$ .

According to the model, is the population of Italy increasing or decreasing? Explain.

Calculation:

Here, we will consider the equation $p=57.563e^{0.0052t}$ .

The function $y=f\left(x\right)$ is said to be an increasing function if $y$ increases as $x$ increases. Here, as the variable $t$ increases then the function $p=57.563e^{0.0052t}$ also increases.

Hence, the population of Italy as per the model given above is $\boxed{Increasing}$.

b.

To determine

To find the populations of Italy in $2000$ and $2012$ .

Answer to Problem 68E

  $\boxed{57.563}$And $\boxed{61.27}$ Millions.

Explanation of Solution

Given information:

The populations (in millions) of Italy from $2000$ through $2012$ can be approximated by the model $p=57.563e^{0.0052t}$ , where $t$ represents the year, with $t=0$ corresponding to $2000$ .

Find the populations of Italy in $2000$ and $2012$ .

Calculation:

Here, the population of Italy in the year $2000$ is got by substituting $t=0$ in the equation. So, if $t=0$ then,

  $p=57.563e^{0.0052\times 0}$

  $p=57.563\times 1$

  $\therefore \left[e^0=1\right]$

  $p=57.563$

Hence, the population of Italy in the year $2000$ is $\boxed{57.563}$Millions.

Now, the population of Italy in the year $2012$ is got by substituting $t=12$ in the equation. So, if $t=12$ then,

  $p=57.563e^{0.0052\times 12}$

  $p=57.563\times 1.0644$

  $p=61.27$

Hence, the population of Italy in the year $2012$ is $\boxed{61.27}$Millions.

c.

To determine

To find the populations of Italy in $2020$ and $2025$ .

Answer to Problem 68E

  $\boxed{63.872}$And $\boxed{65.554}$ Millions.

Explanation of Solution

Given information:

The populations (in millions) of Italy from $2000$ through $2012$ can be approximated by the model $p=57.563e^{0.0052t}$ , where $t$ represents the year, with $t=0$ corresponding to $2000$ .

Use the model to predict the populations of Italy in $2020$ and $2025$ .

Calculation:

Here, the population of Italy in the year $2020$ is got by substituting $t=20$ in the equation. So, if $t=20$ then,

  $p=57.563e^{0.0052\times 20}$

  $p=57.563\times 1.1096$

  $p=63.872$

Hence, the population of Italy in the year $2020$ is $\boxed{63.872}$Millions.

Now, the population of Italy in the year $2025$ is got by substituting $t=25$ in the equation. So, if $t=25$ then,

  $p=57.563e^{0.0052\times 25}$

  $p=57.563\times 1.1388$

  $p=65.554$

Hence, the population of Italy in the year $2025$ is $\boxed{65.554}$Millions.