Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 898. (April 2007)

C. 898. The length of the sides of the regular triangle ABC is 6 cm. A bug starts at the vertex C of the triangle and crawls uniformly towards vertex A at a speed of 4 mm/s. At the same time, another bug starts from vertex B and crawls towards vertex C at a speed of 3 mm/s. How long does it take for the distance between the bugs to reach the smallest value, and what is the smallest distance?

(5 pont)

Deadline expired on May 15, 2007.


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

Megoldás. Az indulástól számított t másodperc múlva legyenek egymástól x távolságra. Ekkor x-re (amennyiben egyik sem ért még célba) felírható:

x^2=(4t)^2+(60-3t)^2-2\cdot4t\cdot(60-3t)\cdot\frac12=
37t^2-600t+3600=37\left[\left(t-\frac{300}{37}\right)^2+\frac{43200}{37^2}\right].

Látható, hogy x a minimumát t=\frac{300}{37}~{\rm sec}\approx 8,11~{\rm sec}-mal az indulás után veszi fel, ekkor egyik bogár sem ért még célba. A minimum értéke:

x=\sqrt{\frac{43200}{37}}~{\rm mm} \approx 34,17~{\rm mm}.


Statistics:

211 students sent a solution.
5 points:96 students.
4 points:29 students.
3 points:14 students.
2 points:3 students.
1 point:9 students.
0 point:45 students.
Unfair, not evaluated:15 solutions.

Problems in Mathematics of KöMaL, April 2007