Solutions to Mathathon

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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 RR, top radius rr and height hh, is V=13πh(R2+r2+Rr).V=\frac{1}{3}\pi h(R^2+r^2+Rr). Therefore we can plug in π227,  h=70,  R=40,  r=70\pi\approx \frac{22}{7}, ~~h=70,~~R=40,~~r=70 to find the volume (must be converted to Liters): V=13×227×70(402+702+40×70=682000(cm3)=682L.V=\frac{1}{3}\times \frac{22}{7}\times 70(40^2+70^2+40\times 70=682000 (\text{cm}^3)=\color{#D61F06}682 \text{L}.

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 xx. By similarity, we get 40x=7070+x\frac{40}{x}=\frac{70}{70+x}. Solve this to get x=2803x=\frac{280}{3}.
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 13πr2h\frac{1}{3}\pi r^2h where rr stands for bottom radius and hh stands for height): Vfrustum=Vbig coneVsmall cone=13π×702×(70+2803)13π×402×2803=682000(cm3)   π is replaced with 13 hereV_{\text{frustum}}=V_{\text{big cone}}-V_{\text{small cone}}=\frac{1}{3}\pi \times 70^2\times (70+\frac{280}{3})-\frac{1}{3}\pi \times 40^2\times \dfrac{280}{3}=682000(\text{cm}^3)~~~\color{#D61F06}\pi~\text{is replaced with}~\frac{1}{3}~\text{here}

3. Use the ‘Baumkuchen’ integral.

If under 18, do NOT do at home without adult supervision to avoid brain explosion.\scriptsize \color{#D61F06} \text{If under 18, do NOT do at home without adult supervision to avoid brain explosion.}
This is mentioned in a book I read which finds volumes of 3D-shapes created by rotating the shape bounded by functions f(x)f(x) and g(x)g(x) in region axba\le x\le b: V=2πabxf(x)g(x)dx.V=2\pi \int_a^b|x||f(x)-g(x)| dx. The absolute value brackets are there to make sure VV 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 Baumkuchen has log-like stripes Explanation:
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 xix_i should be replaced with dxdx :)
For simplicity, here I let g(x)=0g(x)=0. 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 I forgot the absolute value brackets :P Enlarge to see clearer :)
Here the volume of the cuboid is 2πxf(x)2\pi |x||f(x)| because g(x)=0g(x)=0 and the g(x)-g(x) term is therefore neglected. Summing infinite cuboids gives V=2πabxf(x)g(x)dx.V=2\pi \int_a^b|x||f(x)-g(x)| dx. 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 a=0,b=70a=0,b=70, f(x)=70f(x)=70 and g(x)=73x2803g(x)=\frac{7}{3}x-\frac{280}{3}. Plug in these to get V1=2π070x70(73x2803)=2π0704903x73x22×227(2453x279x3070)=75460009.V_1=2\pi\int_0^{70} |x||70-(\frac73x-\frac{280}{3})=2\pi \int_0^{70} \frac{490}{3}x-\frac{7}{3}x^2\approx 2\times\frac{22}{7}(\left. \frac{245}{3}x^2-\frac79x^3\right|_0^{70})=75460009. Similarly we can apply the same to the small cone to get V2=18940009V_2=\frac{1894000}{9}. V1V2=628000.V_1-V_2=628000. Convert to liters: 628628.

Note by Jeff Giff
4 months ago

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You are 13 and u know integration? Now that's CoOoOOOoOlll!

Agent T - 3 months, 3 weeks ago

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Oh uhh yeaaah :)

Jeff Giff - 3 months, 3 weeks ago

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U know what I’m actually making a calculus note to help unsubscribed users like me :)

Jeff Giff - 3 months, 3 weeks ago

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Everything awesome in brilliant is here :)

Jeff Giff - 3 months, 3 weeks ago

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I'm already subbed know! :D

Great job !

Agent T - 3 months, 3 weeks ago

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顺便说一句,因为我年纪大了,所以您可以问我任何与科学或Python(或心理学)有关的问题。我很乐意为您提供帮助:) Hope it made some sense :P

Agent T - 3 months, 2 weeks ago

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@Agent T It’s actually precisely translated I should say :) but I can comprehend English well enough without the help of translators anyway :)

Jeff Giff - 3 months, 2 weeks ago

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@Jeff Giff I wanted to sound CoOoOOOoOlll that's why I did that, not bcz I was doubtful :D

Agent T - 3 months, 2 weeks ago

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@Agent T Besides you might like to know sir Zakir Husain, he sometimes needs help with solving stuff with programs :)

Jeff Giff - 3 months, 2 weeks ago

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@Jeff Giff Yeah sure:)

Agent T - 3 months, 2 weeks ago

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You know what?
1. I joined brilliant one year ago, when I lacked at calculus :P (Besides brilliant doesn’t allow me to chose 12-year-old! I changed it back on my birthday)
2. You’re the only one to realise my age
3. I am the youngest brilliant user known for the time being
4. I feel like Sheldon

Jeff Giff - 3 months, 3 weeks ago

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1.Haha what a fun story ,the best part was u telling that u were not so good in calculus at the time when I didn't even know what the heck it is XP.

2.Are u serious?I mean it's mentioned in almost all the ques posted by u¯\(ツ)/¯.

3.I didn't wanna say it but.....

Aww

.4. Haha ,weird flex but okay.. [face-palm emoji] ᕙ( • ‿ • )ᕗ

Agent T - 3 months, 3 weeks ago

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@Agent T Haha :D

Jeff Giff - 3 months, 2 weeks ago

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@Agent T Well, people have to go to my profile to see my age so yeah, hardly anyone knows!

Jeff Giff - 3 months, 2 weeks ago

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@Agent T Actually some dude in America does calculus better than me (screw it! :P) and he’s only 15 :D besides integration always gives me a headache (goddamn calculus fries my brain every time ¯(ツ)/¯)

Jeff Giff - 3 months, 2 weeks ago

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@Jeff Giff Haha it's okay,you will get the essence of it with lots and lots of practice.Good luck:)

Agent T - 3 months, 2 weeks ago

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@Agent T Now coming to think of it... I suck at physics. Perhaps mastering physics will be my goal after mastering calculus :)

Jeff Giff - 3 months, 2 weeks ago

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@Jeff Giff I loOoOve physics,ask me anything and I'll spill all the facts about that topic:D

Agent T - 3 months, 2 weeks ago

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