Answer
See Below
Work Step by Step
(a)
Consider the equation
\[B=1.7{{x}^{2}}+6x+26\]
Consider that \[x=0\] corresponds to the year 2003 then to determine the number of bicycle-friendly communities in the year 2011.
Put \[x=8\] as it corresponds to the year 2011 in the equation \[B=1.7{{x}^{2}}+6x+26\] and simplify it as follows:
\[\begin{align}
& B=1.7{{\left( 8 \right)}^{2}}+6\left( 8 \right)+26 \\
& =182.8
\end{align}\]
By rounding to the closest integer, the number of bicycle-friendly communities in the year 2011 is \[183\].
Now, in the above graph observe that the value 183 is anoverestimate by 3 as the number in the graph corresponding to the year 2011 is 180.
(b)
Consider the equation
\[B=1.7{{x}^{2}}+6x+26\]
Now,to find the year in which 826 U.S. communities will be bicycle friendly, put \[B=826\]in the above equation
\[\begin{align}
& B=1.7{{x}^{2}}+6x+26 \\
& 826=1.7{{x}^{2}}+6x+26
\end{align}\]
Subtract 26 from both the sides of the above equation
\[1.7{{x}^{2}}+6x-800=0\]
Compare this equation with the equation
\[a{{x}^{2}}+bx+c=0\], where\[a=1.7,\ b=6,\text{ and }c=-800\]. Now, put these values in the quadratic formula
\[\begin{align}
& x=\frac{-6\pm \sqrt{{{\left( 6 \right)}^{2}}-4\times \left( 1.7 \right)\times \left( -800 \right)}}{2\times \left( 1.7 \right)} \\
& =\frac{-6\pm \sqrt{36+5440}}{2} \\
& =\frac{-6\pm \sqrt{5476}}{2} \\
& =\frac{-6\pm 74}{2}
\end{align}\]
Further simplifying
\[\begin{align}
& x=\frac{-6\pm 74}{3.4} \\
& =\frac{-6+74}{3.4},\frac{-6-74}{3.4} \\
& =\frac{68}{3.4},\frac{-80}{3.4} \\
& =20,-23.5
\end{align}\]
Since, \[x\] cannot be negative therefore, \[x=20\].
So, the required year is \[2003+20=2023\].
