A mathematician is buying a Christmas tree from Revolutionary Christmas Trees.
He sends this model function ( x in terms of y ):
x = ⎩ ⎪ ⎨ ⎪ ⎧ 2 − ( 2 ⌊ y ⌋ + { y } ) 2 1 0 0 ≤ y < 3 − 1 ≤ y < 0 otherwise
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 6 4 0 π .
Find the value of s .
Note: The notation { y } means the fractional part of y
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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 gives a cylindrical stump of volume 4 π .
The second layer from 0 ≤ y < 1 joined with the fourth layer from 2 ≤ y < 3 gives a cone of volume 3 1 ( 8 π ) .
The third layer 1 ≤ y < 2 is a frustum of volume 3 1 ( 4 1 3 π ) .
Adding these all together gives a total volume of 4 π , so x = 4 π 6 4 0 π = 1 6 0 .
Frankly speaking I don't see any calculus here.
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Yes, that's why I tagged the problem under geometry as well. :)
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oh yeah...
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
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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 :)
When you calculate volume of solid as 4 π but calculate 4 π 6 4 0 π as 1 6 . Bad luck! Corrected in second attempt!
Pro tip to help graph the function, just replace y with x and the function can be manipulated to 2 − ( 2 x + x )
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Calculus: