Chapter 6.3, Problem 22E

a.

To determine

To compare:The population of both towns.

Answer to Problem 22E

The population in Town $A$ grows exponentially than the population in Town $B$ .

Explanation of Solution

Given:

Population of town $A$ in $1970$ was $3000$ . Population of town $A$ increased by $20\%$ every decade. The graph for population of town $B$ given.

Concept Used:

Use patterns between consecutive data pairs to determine type of function. Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Calculation:

Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Graph given

  

We determine the population of town $A$ as a function of the number of decades after $1970$ :

  $\begin{array}{l}f(x)=a{b}^x\\ a=3000\\ b=1+0.2=1.2\\ $ f(x)=3000{(1.2)}^x$\end{array}$

We determine the rate of change from $1990$ to $2010$

Town $A$

  $\frac{f(4)-f(2)}{4-2}=\frac{3000{(1.2)}^4-3000{(1.2)}^2}{2}\approx 950.4$

Town $B$

  $\frac{g(4)-g(2)}{4-2}=\frac{6250-5000}{2}=625$

The population in Town $A$ grows exponentially and even if it starts from 3000 instead of 4000 as the population in Town $B$ , it soon gets above it:

  $\begin{array}{l}$ f(4)=3000{(1.2)}^4\approx 6221$g(4)=6250\\ $ f(5)=3000{(1.2)}^5\approx 7465$g(5)=6500\end{array}$

Hence,

The population in Town $A$ grows exponentially than the population in Town $B$ .

b.

To determine

To find: The town will have greater population after $2020$ .

Answer to Problem 22E

After 5 decades, the population of Town $A$ will be higher than the population in Town $B$ .

Explanation of Solution

Given:

Population of town $A$ in $1970$ was $3000$ . Population of town $A$ increased by $20\%$ every decade. The graph for population of town $B$ given.

Concept Used:

Use patterns between consecutive data pairs to determine type of function. Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Calculation:

Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Graph given

  

The population in Town $A$ grows exponentially and even if it starts from 3000 instead of 4000 as the population in Town $B$ , it soon gets above it:

  $\begin{array}{l}$ f(4)=3000{(1.2)}^4\approx 6221$g(4)=6250\\ $ f(5)=3000{(1.2)}^5\approx 7465$g(5)=6500\end{array}$

So, after 5 decades, the population of Town $A$ will be higher than the population in Town $B$ . Five decades correspond to the year $1970+5(100)=2020$ .

Hence,

So, after 5 decades, the population of Town $A$ will be higher than the population in Town $B$ .