Day 6: Revolutionary Christmas Trees

Calculus Level 2

A mathematician is buying a Christmas tree from Revolutionary Christmas Trees.

He sends this model function ( xx in terms of yy ):

x={2(y2+{y})0y<3121y<00otherwise 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 yy-axis to create a solid of revolution to model his tree.

He then sends a volume enlargement scale factor ss by which the volume of the solid is multiplied to make the tree the correct size. He wishes to have a final volume of 640π640 \pi .

Find the value of ss.

Note: The notation {yy} means the fractional part of yy


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