Day 6: Revolutionary Christmas Trees

Calculus Level 4

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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