Day 6: Revolutionary Christmas Trees

Calculus Level 2

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

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

x = { 2 ( y 2 + { y } ) 0 y < 3 1 2 1 y < 0 0 otherwise 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 y -axis to create a solid of revolution to model his tree.

He then sends a volume enlargement scale factor s 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 π 640 \pi .

Find the value of s s .

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


The answer is 160.

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2 solutions

Calculus:

Michael Ng
Dec 5, 2014

Plotting the function gives half of a Christmas tree shape. Now we consider the layers of the Christmas tree when they are rotated.

The first layer from 1 y < 0 -1 \leq y < 0 gives a cylindrical stump of volume π 4 \frac{\pi}{4} .

The second layer from 0 y < 1 0 \leq y < 1 joined with the fourth layer from 2 y < 3 2 \leq y < 3 gives a cone of volume 1 3 ( 8 π ) \frac{1}{3}(8 \pi) .

The third layer 1 y < 2 1 \leq y < 2 is a frustum of volume 1 3 ( 13 4 π ) \frac{1}{3}(\frac{13}{4}\pi) .

Adding these all together gives a total volume of 4 π 4 \pi , so x = 640 π 4 π = 160 x = \frac{640 \pi}{4 \pi} = \boxed{160} .

Frankly speaking I don't see any calculus here.

Julian Poon - 6 years, 6 months ago

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Yes, that's why I tagged the problem under geometry as well. :)

Michael Ng - 6 years, 6 months ago

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oh yeah...

Julian Poon - 6 years, 6 months ago

Correct me if I'm wrong, but isn't it volume factor: not scale factor, since scale factor is the cube root of the volume factor

Trevor Arashiro - 6 years, 6 months ago

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@Trevor Arashiro Yes, I did think about that, but volume enlargement scale factor might sound cumbersome so I added an explanation. I'll change it anyway as it is more correct. Thank you :)

Michael Ng - 6 years, 6 months ago

When you calculate volume of solid as 4 π 4\pi but calculate 640 π 4 π \dfrac{640\pi}{4\pi} as 16 16 . Bad luck! Corrected in second attempt!

Pranjal Jain - 6 years, 6 months ago

Pro tip to help graph the function, just replace y with x and the function can be manipulated to 2 ( x + x 2 ) 2-(\frac{x+{x}}{2})

Trevor Arashiro - 6 years, 6 months ago

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