Answer
See Below
Work Step by Step
(a)
The mathematical models given represent the data between the p and x, where p represents percentage of American adults who reported personal gun ownership after
x years of 1980 till 2010
Consider the given model, the graph of percentage of American adults reporting personal gun ownership.
For the graph given, the value of p, that is, percentage reporting personal gun ownership is 20.8
The variation of the percentage ownership with respect to the time (in years), is given by the mathematical model, using (I),
\[p=-0.3x+30\] (II)
The value of x is 30
From (II),
\[\begin{align}
& p=-0.3x+30 \\
& p=-0.3\left( 30 \right)+30 \\
& p=-9+30 \\
& p=21
\end{align}\]
The value of percentage reporting personal gun ownership is 21, which underestimates the percentage shown by the graph by the value of difference between the two values,
\[21-20.8=0.2\]
Hence, the percentage of American adults reporting personal gun ownership in 2010 according to the mathematical model is 21,underestimates the percentage shown by the given graph by 0.2, for the value 20.8
(b)
The mathematical models given represent the data between the p and x, where p represents percentage of American adults who reported personal gun ownership after
x years of 1980 till 2010
Consider the model 1, the graph of percentage of American adults reporting personal gun ownership.
For the graph given, the value of p, that is, percentage reporting personal gun ownership is 20.8
The variation of the percentage ownership with respect to the time (in years), is given by the mathematical model, using (I),
\[p=-0.003{{x}^{2}}-0.22x+30\] (II)
The value of x is 30
From (II),
\[\begin{align}
& p=-0.003{{x}^{2}}-0.22x+30 \\
& p=-0.003{{\left( 30 \right)}^{2}}-0.22\left( 30 \right)+30 \\
& p=-0.003\left( 30\times 30 \right)-6.6+30 \\
& p=-2.7+23.4
\end{align}\]
Simplifying it further, to get,
\[p=20.7\]
The value of percentage reporting personal gun ownership is \[20.7\], which overestimates the percentage shown by the graph by the value of difference between the two values,
\[20.8-20.7=0.1\]
Hence, the percentage of American adults reporting personal gun ownership in 2010 according to the mathematical model is 20.7,overestimates the percentage shown by the given graph by 0.1, for the value 20.8.
(c)
The mathematical models given represent the data between the p and x, where p represents percentage of American adults who reported personal gun ownership after
x years of 1980 till 2010
Consider the given model, the graph of percentage of American adults reporting personal gun ownership.
For the graph given, the value of p, that is, percentage reporting personal gun ownership is \[177%\].
The variation of the percentage ownership with respect to the time (in years), is given by the mathematical model, using (I),
\[p=-0.3x+30\] (II)
The value of p is 177
From (II),
\[\begin{align}
& p=-0.3x+30 \\
& 177=-0.3x+30 \\
& 0.3x=-177+30 \\
& x=-\frac{147}{0.3}
\end{align}\]
\[\begin{align}
& \ \ \ \ \,x=-\frac{147}{0.3} \\
& x=490
\end{align}\]
The value of x, that is, the time (in years) is 490 years. For the year in which the percentage becomes \[177%\], the value of x is added to 1980 to get,
\[1980+490=2470\]
