Volume of a huge snowball

Dec 2019
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The ten Christmas elves Elisa, Frida, Gustav, Heinrich, Ida, Johann, Karla, Ludwig, Marta, and Norwin had a lot of work to do during the past weeks: they tinkered, painted, and sawed. But now, all presents are wrapped. Proud of their work, they high-fived each other—when Elisa looks out of the window and notices, "It is snowing a lot!"
"Let's go outside and have a snowball fight!" Gustav says eagerly. Our ten Christmas elves put on warm clothes, run outside, and have a long and merry snowball fight.

At some point, Ida notices, "Folks! This spherical snowball fits perfectly into the snow-free rain gutter. The snowball touches each of the rain gutter's edges at exactly one point and lines up precisely with its top."
This amazing discovery causes the Christmas elves to abruptly end their snowball fight. Since they do not only have a distinct sense of delightful Christmas presents, but also a deep love for Mathematics, they immediately want to calculate the snowball's volume. To this end, they use the following figure showing the cross section of the rain gutter and snowball (not to scale)

index.png

The elves begin to calculate and come up with different formulas.
Who is right? Give your calculations.

  1. After a little while, Elisa calculates:


  2. Frida needs a bit more time, but finally she states the following result:


  3. Gustav puzzles over the problem patiently and is confident about his result:


  4. Heinrich is very excited about his result, since he solved such a problem not long ago. He quickly obtains:


  5. Ida is very concentrated while calculating. In the end, she gets:


  6. Johann is very absorbed, smiles, and states his result:


  7. Karla follows a different approach and obtains the following volume formula:


  8. Ludwig is very passionate about this task; so he makes his own optimized sketch and is able to present the following result using geometrical and analytical approaches:


  9. Marta trusts in her mathematical skills and is surprised about the volume she calculated:


  10. Norwin leans back, looks at the sketch for quite a long time, and says, "Without further information, one cannot solve this problem!"
 

skipjack

Forum Staff
Dec 2006
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I used Pythagoras and this formula to obtain the radius of the snowball, and then the usual formula for the volume of a sphere in terms of its radius.
 
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Dec 2019
52
1
ok
I used Pythagoras and this formula to obtain the radius of the snowball, and then the usual formula for the volume of a sphere in terms of its radius.
Can you explain it to me step by step just like you did with the macaroons thingie? You're like the smartest person and you're good at explaining.
 

skipjack

Forum Staff
Dec 2006
21,301
2,377
The cross-section of the gutter and the line across the top of it form a right-angled triangle with legs of length $\sqrt{2}$dm (by Pythagoras) and hypotenuse of length $2\hspace{1px}$dm, so the formula I linked gives the radius of its incircle as $(\sqrt2 - 1)$dm. This is the radius of the snowball, so its volume is $\frac43\pi(\sqrt2 - 1)^3\hspace{1px}\text{dm}^3 = \frac43\pi(5\sqrt2 - 7)\hspace{1px}\text{dm}^3$.