, 2014

The song’s position on week 1 ($w=1$), is $P(1)=3(1)^2-36(1)+110=77$. The song’s position is given by the quadratic function $P=3w^2-36w+110$, the graph of which is a parabola opening upward. The minimum value of this parabola is achieved at its vertex. To find the coordinates of the vertex, we may complete the square. $$\begin{align*} P& =3w^2-36w+110\\ & =3(w^2-12w)+110\\ & =3(w^2-12w+36-36)+110\\ & =3(w^2-12w+36)-108+110\\ & =3(w-6)^2+2\end{align*}$$ Therefore, the vertex of the parabola occurs at $w=6$ and $P=2$. The best position that the song “Recursive Case" reaches is position #2. The song reaches its best position on week 6. To determine the last week that “Recursive Case" appears on the Top 200 list, we want to find the largest $w$ such that $P=3w^2-36w+110\leq200$. Using the vertex form from part (b), we have $3(w-6)^2+2\leq200$ or $3(w-6)^2\leq198$ or $(w-6)^2\leq66$. To determine the largest positive integer $w$ such that $(w-6)^2\leq66$, we want to find the largest square that is less than or equal to 66. Since $8^2\leq66$ and $9^2>66$, then the largest $w$ satisfies $w-6=8$ and so $w=14$. The last week that “Recursive Case" appears on the Top 200 list is week 14. To check this we note that, $$\text{P(14)=3(14-6) 2+2=194 200 but P(15)=3(15-6) 2+2=245>200.}$$