# Volume of a huge snowball

#### absoluzation

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)

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
You can trust an elf who smiles.

topsquark

#### absoluzation

You can trust an elf who smiles.
Haha, I love your humor. Could you tell me how to do it tho cuz I didn't think it was Johann?

#### skipjack

Forum Staff
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.

topsquark

#### absoluzation

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.
I can't see what formula you're talking about

#### absoluzation

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.

#### absoluzation

Nvm, I found out myself. THANKS SO MUCH AGAIN!

#### skipjack

Forum Staff
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 $\frac43\pi(\sqrt2 - 1)^3\hspace{1px}\text{dm}^3 = \frac43\pi(5\sqrt2 - 7)\hspace{1px}\text{dm}^3$.