# Day 6: Revolutionary Christmas Trees

Calculus Level 3

A mathematician is buying a Christmas tree from 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 $$y$$

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