Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 275. (December 2010)

K. 275. A flying saucer with aliens on board is travelling at a uniform altitude above the ground (that is, at a constant distance from the Earth) at a uniform speed of 800 km/h. At 8 a.m., it was over London, and after 1 hour and 24 minutes it is already over Berlin. Assume that the Earth is a sphere of radius 6370 km. The distance between London and Berlin is 929 km measured along the surface. The saucer travels the shortest path between the two points under the given conditions. Calculate its distance from the surface of the Earth.

(6 pont)

Deadline expired on January 10, 2011.


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

Megoldás. Ha a csészelj a Föld felszínétől \(\displaystyle h\) távolságra repül - a feltételek szerint a Londont Berlinnel összekötő főkörívvel koncentrikus és hasonló körív mentén. Ezért 1h24perc, azaz 1,4 óra alatt \(\displaystyle (6370+h)\cdot \frac{929}{6370}\)km-t tesz meg, így a megtett útjára a következő összefüggés igaz: \(\displaystyle (6370+h)\cdot \frac{929}{6370}=1,4\cdot 800\). Átalakítás után \(\displaystyle h\approx 1309,656\). A csészealj kb. 1309,656km-re van a Föld felszínétől.


Statistics:

198 students sent a solution.
6 points:164 students.
5 points:11 students.
4 points:4 students.
3 points:1 student.
2 points:6 students.
1 point:4 students.
0 point:3 students.
Unfair, not evaluated:5 solutions.

Problems in Mathematics of KöMaL, December 2010