Mathematical and Physical Journal
for High Schools
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Problem C. 869. (October 2006)

C. 869. A cylinder of height \frac{4}{3} R is inscribed into a sphere of radius R. Determine the ratio of their volumes.

(5 pont)

Deadline expired on November 15, 2006.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Jelöljük R-rel a gömb, r-rel a henger sugarát, és m-mel a henger magasságát.

Ekkor az ábra alapján a Pitagorasz-tételt felírva:

(2R)^2 = (2r)^2 + (\frac43R)^2,

ahonnan

r^2 = \frac59R^2.

A térfogatok aránya innen

\frac{V_{\rm{henger}}}{V_{\rm{g\"omb}}} = \frac{r^2 \pi m}{\frac{4R^3 \pi}3} = \frac{\frac{5R^2 \pi}9 \cdot \frac43R}{\frac{4R^3 \pi}3}  = \frac59.

Tehát a henger a gömb térfogatának \frac59-része.


Statistics:

570 students sent a solution.
5 points:451 students.
4 points:53 students.
3 points:18 students.
2 points:23 students.
1 point:6 students.
0 point:11 students.
Unfair, not evaluated:8 solutions.

Problems in Mathematics of KöMaL, October 2006