Chapter 3, Problem 15PS

a.

To determine

To calculate: The exponential model for the data.

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Answer to Problem 15PS

The exponential model is $y=252602.75{\left(1.36\right)}^t$ .

Explanation of Solution

Given information:

The population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Calculation:

Consider the population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

It is provided that y represent the population in the year t with $t=0$ that corresponds to 1700.

Follow the steps provided below to estimate the exponential model.

Step 1: Press $\text{STAT}$ key.

Step 2: Press $\text{ENTER}$ key.

Step 3: Enter the column 1 of table provided above in $\text{L1}$ and press $\text{ENTER}$ key.

Step 4: Enter the column 2 of table provided above in $\text{L2}$ and press $\text{ENTER}$ key.

Step 5: Press $\text{2nd}$ key and $\text{MODE}$ key.

Step 6: Press $\text{STAT}$ key, select the $\text{CALC}$ menu.

Step 7: Select the fourth option that is $\text{ExpReg}$ and press $\text{ENTER}$ key.

Step 8: Press $\text{2nd}$ key and press 1 to call list $\text{L1}$ and put comma.

Step 9: Press $\text{2nd}$ key and press 2 to call list $\text{L2}$ and press $\text{ENTER}$ key.

The result obtained on screen is provided below.

  

Therefore, exponential model is,

  $\begin{array}{c}y=a{b}^x\\ y=252602.75{\left(1.36\right)}^t\end{array}$

b.

To determine

To calculate: The quadratic model for the data.

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Answer to Problem 15PS

The quadratic model is $y=40090.152t^2-14649.545t+291776.97$ .

Explanation of Solution

Given information:

The population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Calculation:

Consider the population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

It is provided that y represent the population in the year t with $t=0$ that corresponds to 1700.

Follow the steps provided below to estimate the quadratic model.

Step 1: Press $\text{STAT}$ key.

Step 2: Press $\text{ENTER}$ key.

Step 3: Enter the column 1 of table provided above in $\text{L1}$ and press $\text{ENTER}$ key.

Step 4: Enter the column 2 of table provided above in $\text{L2}$ and press $\text{ENTER}$ key.

Step 5: Press $\text{2nd}$ key and $\text{MODE}$ key.

Step 6: Press $\text{STAT}$ key, select the $\text{CALC}$ menu.

Step 7: Select the fourth option that is $\text{ExpReg}$ and press $\text{ENTER}$ key.

Step 8: Press $\text{2nd}$ key and press 1 to call list $\text{L1}$ and put comma.

Step 9: Press $\text{2nd}$ key and press 2 to call list $\text{L2}$ and press $\text{ENTER}$ key.

The result obtained on screen is provided below.

  

Therefore, quadratic model is,

  $\begin{array}{c}y=a{x}^2+bx+c\\ y=40090.152t^2-14649.545t+291776.97\end{array}$

c.

To determine

To graph: The models for the data.

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Explanation of Solution

Given information:

The population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Graph:

Consider the population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

The exponential model is $y=252602.75{\left(1.36\right)}^t$ .

The quadratic model is $y=40090.152t^2-14649.545t+291776.97$ .

Plot the data points and all the models in the coordinate plane.

  

The green curve line represents $y=252602.75{\left(1.36\right)}^t$ , red curve represents $y=40090.152t^2-14649.545t+291776.97$ black curve represents the data.

Interpretation:

All the models and the data points itself coincide with each other. All the models fits with the data.

d.

To determine

To explain: The model that best fits the. Also the model that best fits the data can be used to predict the population of United Sates in 2018.

Answer to Problem 15PS

The model that best fits the data is exponential model. No, this cannot be used to predict the population.

Explanation of Solution

Given information:

The population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

Calculation:

Consider the population estimates of the American colonies from 1700 to 1780 is provided below,

  $\begin{array}{cc}\text{Year}& \text{Population}\\ 1700& 250,900\\ 1710& 331,700\\ 1720& 466,200\\ 1730& 629,400\\ 1740& 905,600\\ 1750& 1,170,800\\ 1760& 1,593,600\\ 1770& 2,148,400\\ 1780& 2,780,400\end{array}$

The quadratic model is $y=40090.152t^2-14649.545t+291776.97$ .

The exponential model is $y=252602.75{\left(1.36\right)}^t$ .

From the graphs it is clear that exponential model coincide with actual data points.

Exponential model cannot be used to predict the population as exponential function is increasing function and values obtained would be very high.

There would be a very high growth rate.

Thus, the model that best fits the data is exponential model. No, this cannot be used to predict the population.