Answer
See below:
Work Step by Step
(a)
Consider the given graph.
The above graph shows that the number of people has a rapid change with the increase in year. If we plot a graph for the value provided, we will get a curve that has low start but very high end.
The exponential plot on the graph has low start and a very high end point. This is a similar increase to the one shown in the bar graph.
Thus, this shows that exponential function has to be used to model the population data given in graph.
An exponential function has to be used to model the population data given in graph.
(b)
Now, simplify the model as
$y=3.476{{\left( 1.023 \right)}^{x}}$
This can be expressed on the graphic calculator as shown below:
\[{{Y}_{\text{1}}}\text{=3}\text{.476*}\left( \text{1}\text{.023} \right)\text{ }\!\!\hat{\ }\!\!\text{ }x\]
Thus, the model is expressed on the graphic calculator as shown above.
Hence, the model is expressed on the graphic calculator as explained above.
(c)
Consider the model shown in the above figure.
$y=3.476{{\left( 1.023 \right)}^{x}}$
Substitute the value of x as 111 for the year 2010, because x represents the year after 1899. It is substituted as shown below:
$\begin{align}
& y=3.476{{\left( 1.023 \right)}^{111}} \\
& =43.37855788 \\
& =43.4
\end{align}$
The population shown by the bar graph in the year 2010 is 44.6 million. This shows that the calculated value using the model underestimates the value shown in the bar graph.
Now, subtract the calculated value from the value obtained from the bar graph for the year 2010.
\[44.6-43.4=1.2\]
Thus, the calculated value underestimates the value from the bar graph by 1.5 million for the year 2010.
Hence, the calculated number is 43.4 million and this value underestimates the value from the bar graph by 1.2 million for the year 2010.
(d)
Consider the model shown in the above figure.
$y=3.476{{\left( 1.023 \right)}^{x}}$
Substitute the value of x as 121 for the year 2020, because x represents the year after 1899. It is substituted as shown below:
$\begin{align}
& y=3.476{{\left( 1.023 \right)}^{121}} \\
& =54.454208 \\
& =54.5\text{ million}
\end{align}$
The population shown by the bar graph in the year 2020 is 53.7 millions. This shows that the calculated value using the model overestimates the value shown in the bar graph.
Now, subtract the calculated value from the value obtained from the bar graph for the year 2020.
$54.5-53.7=0.8\text{ }$
Thus, the calculated value overestimates the value from the bar graph by 0.8 million for the year 2020.
Hence, the calculated number is 54.5million and it overestimates the value from the bar graph by 0.8 million for the year 2020.
