I LOVE THE DREAM SMP!
There are many ways to look at this problem. Three good ways would be:
1. Directly calculate if you know the formula for frustum volume.
The formula for frustum volume, given bottom radius , top radius and height , is
Therefore we can plug in to find the volume (must be converted to Liters):
2. Calculate difference of cones not knowing the formula for frustum volume.
Extending the cone by imagination and using similarity, we get the pic below:
Let the height of the increased cone be . By similarity, we get . Solve this to get .
So the volume of the frustum is the volume of the larger cone minus the volume of the smaller cone (the formula for cone volume is where stands for bottom radius and stands for height):
3. Use the ‘Baumkuchen’ integral.
This is mentioned in a book I read which finds volumes of 3D-shapes created by rotating the shape bounded by functions and in region :
The absolute value brackets are there to make sure is positive. THIS WORKS ONLY IF THE ROTATED BOUND IS ON THE SAME SIDE OF THE Y-AXIS.
Why is it called Baumkuchen?
It is because Baumkuchen is ‘log-like dessert’ in German, and the integral is like one!
Baumkuchen has log-like stripes
We can split the integral into rings formed by the original bound split and rotated around the y-axis individually.
Here I made a little mistake when labelling :P should be replaced with :)
For simplicity, here I let . Of course this can be generalised to other functions as well, given above.
If we cut a single ‘ring’ open, we get its volume by seeing it as a cuboid:
I forgot the absolute value brackets :P
Enlarge to see clearer :)
Here the volume of the cuboid is because and the term is therefore neglected. Summing infinite cuboids gives
So we can see the frustum as a cone chopped off from another as in example 2.
This way, the volume of the big cone is the integral with , and .
Plug in these to get
Similarly we can apply the same to the small cone to get .
Convert to liters: .