Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 1188. (October 2013)

C. 1188. A circular sector is folded to form a conical hat. What is the central angle of the sector if the height of the hat equals four fifths of the radius of the sector?

(5 pont)

Deadline expired on November 11, 2013.


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

Megoldás. Jelölje a kúp alkotóját \(\displaystyle a\), alapkörének sugarát \(\displaystyle r\), a körcikk középponti szögét pedig \(\displaystyle \varphi\). Tudjuk, hogy a kúp magassága \(\displaystyle \frac45a\). A kúp alkotójára, magasságára és sugarára felírható Pitagorasz tétele:

\(\displaystyle a^2=r^2+\left(\frac45a\right)^2,\)

amiből \(\displaystyle r=\frac35a\).

A kúppalást területe \(\displaystyle \pi ra\), a körcikk területe \(\displaystyle \frac{\varphi}{2\pi}\cdot a^2\pi=\varphi\cdot\frac{a^2}{2}\). A kettő egyenlő, vagyis

\(\displaystyle \pi ra=\varphi\cdot\frac{a^2}{2},\)

amiből

\(\displaystyle \varphi=\frac{2\pi ra}{a^2}=\frac{2\pi\cdot\frac35a}{a}=\frac65\pi.\)


Statistics:

94 students sent a solution.
5 points:72 students.
4 points:5 students.
3 points:3 students.
2 points:7 students.
1 point:2 students.
0 point:4 students.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, October 2013