# Day 6: Revolutionary Christmas Trees

**Calculus**Level 4

*Revolutionary*Christmas Trees.

He sends this model function ( \(x\) in terms of \(y\) ):

\[ x = \begin{cases} 2 - ( \frac{\lfloor y \rfloor }{2} + \{ y \} ) & 0 \leq y < 3\\ \frac{1}{2} & -1 \leq y < 0\\ 0 & \text{otherwise} \end{cases} \]

This is rotated round the \(y\)-axis to create a solid of revolution to model his tree.

He then sends a volume enlargement scale factor \(s\) by which the volume of the solid is multiplied to make the tree the correct size. He wishes to have a final volume of \(640 \pi \).

Find the value of \(s\).

**Note:** The notation {\(y\)} means the fractional part of \(y\)