Answer
a) $183$ communities
b) $2023$
Work Step by Step
a) The given formula for the number of bicycle-friendly communities $B$, $x$ years after $2003$ is $$B=1.7x^2+6x+26.$$ This means $2003$ as a starting year will be taken as 0th year and $2004$, the year after $2003$ will be the 1st year and so on. Therefore, the year $2011$ will be the 8th year after $2003$ and $x$ will be equal to $8$ for $2011$.
Applying the formula for $x$ equals $8$, $$B=1.7(8^2)+6(8)+26=1.7(64)+6(8)+26=182.8$$ rounding solution the nearest whole number will result in $B$ $=$ $183$ and our rounded value will overestimate the number shown by the graph.
b) To find the year the in which the $826$ U.S. communities will be bicycle friendly, we will set the number of bicycle friendly communities to $826$ and find the $x$. $$826=1.7x^2+6x+26$$ then substract $826$ from both sides, $$826-826=1.7x^2+6x+26-826$$ $$1.7x^2+6x-800=0$$ and the applying the quadratic formula we get: $$x=\frac{-6\pm\sqrt{6^2-4(1.7)(-800)}}{2(1.7)}$$ so $$x_1\approx -23.5,x_2=20.$$ Since $x$ is a number of years it has to be positive. Therefore, the year our bicycle friendly communities will equal $826$ will be $$2003+20=2023.$$
