Chapter 8.2, Problem 38PPS

(a)

To determine

To find: the equation for population growth model in terms of $x$.

Answer to Problem 38PPS

The equation for the area $a$ defended by beaver weighing $w\text{pounds}$ is $y=1.161w^{1.31}$

Explanation of Solution

Given information:

In 1950, the world population was about $2.556\text{billion}$ and by 1980, it had increased to about $4.458\text{billion}$.

Consider the given information.

In 1950, the world population was about $2.556\text{billion}$ and by 1980, it had increased to about $4.458\text{billion}$.

So, In the $30\text{years}$ , the population are increased from $2.556\text{billion}$ to $4.458\text{billion}$.

Substitute $4.458$ for $y$ , $2.556$ for $a$ , and $30$ for $x$ in the general exponential function $y=a{b}^x$ ..

  $\begin{array}{c}4.458=2.556{\left(b\right)}^x\\ b^{30}=\frac{4.458}{2.556}\\ b={\left(1.744\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.}\\ =1.019\end{array}$

Substitute $2.556$ for $a$ , a $1.019$ for $b$ in the equation $y=a{b}^x$.

  $y\left(x\right)=2.556{\left(1.019\right)}^x$

Therefore, the equation for population growth model in terms of $x$ is $y\left(x\right)=2.556{\left(1.019\right)}^x$.

(b)

To determine

To find: the population of the world in the year 2000.

Answer to Problem 38PPS

The population in the year 2000 is $6.55\text{billion}$.

Explanation of Solution

Given information:

In 1950, the world population was about $2.556\text{billion}$ and by 1980, it had increased to about $4.458\text{billion}$.

Consider the given information.

Substitute $50$ for $x$ in the exponential function $y\left(x\right)=2.556{\left(1.019\right)}^x$.

  $\begin{array}{c}y\left(x\right)=2.556{\left(1.019\right)}^{50}\\ =2.556\left(2.56\right)\\ =6.55\text{billion}\end{array}$

Therefore, the population in the year 2000 is $6.55\text{billion}$.

(c)

To determine

To find: the comparison of the estimated and actual population result.

Answer to Problem 38PPS

The estimate population of the world is greater than the actual population of the world.

Explanation of Solution

Given information:

In 1950, the world population was about $2.556\text{billion}$ and by 1980, it had increased to about $4.458\text{billion}$.

Consider the given information.

The calculated population of the world is $6.55\text{billion}$ and exact population of the world is $6.08\text{billion}$.

So, the estimate population of the world is greater than the actual population of the world.

Therefore, the estimate population of the world is greater than the actual population of the world.

(d)

To determine

To find: the population of the world in the year 2020.

Answer to Problem 38PPS

The population in the year 2020 is $9.55\text{billion}$.

Explanation of Solution

Given information:

In 1950, the world population was about $2.556\text{billion}$ and by 1980, it had increased to about $4.458\text{billion}$.

Consider the given information.

Substitute $70$ for $x$ in the exponential function $y\left(x\right)=2.556{\left(1.019\right)}^x$.

  $\begin{array}{c}y\left(70\right)=2.556{\left(1.019\right)}^{70}\\ =2.556\left(3.73\right)\\ =9.55\text{billion}\end{array}$

Therefore, the population in the year 2020 is $9.55\text{billion}$.