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K. 55. There are two milk bars in Cowton, and both of them have hot chocolate with frothed chocolate topping on their menus. Each milk bar serves in a cylindrical glass of the same height. (On serving, one half of the volume of the drink is liquid chocolate, and the other half is frothed chocolate.) In a short time, the froth turns into liquid chocolate of one quarter as much volume. In the Jolly Cowboy, chocolate is served in glasses of radius 6 cm, and sold for 12 Cowton cents a glass. In the Happy Cowgirl, the radius of the glasses is 5 cm, but they fill up the glass again when the froth of the first filling has settled. They charge 11 Cowton cents for a glass. In which milk bar is chocolate cheaper? [To obtain the volume of a cylinder, the area of its base is multiplied by its height.] (Based on a problem of the 24th József Öveges Memorial Competition)

(6 points)

This problem is for grade 9 students only.

Deadline expired on 12 December 2005.

Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldás: A Víg Tehénpásztorban a bögre térfogatának {1\over2}+{1\over2}\cdot{1\over4}={5\over8} része a kapott kakaó mennyisége. A Jókedvű Csordásban a bögre térfogatának {5\over8}+{1\over2}\cdot{3\over8}+{1\over2}\cdot{3\over8}\cdot{1\over4}={55\over64} része a kapott kakaó mennyisége. Legyen a bögrék magassága h cm! A Víg Tehénpásztorban a bögre térfogata köbcentiméterben mérve 36\pi.h, a kakaó mennyisége pedig 36\pi\cdot h\cdot{5\over8}={45\over2}\cdot\pi\cdot h, egy petákért \left({45\over2}\cdot\pi\cdot h\right):12={15\over8}\cdot\pi\cdot h köbcenti jár. A Jókedvű Csordásban a bögre térfogata köbcentiméterben mérve 25\pi.h, a kakaó mennyisége pedig 25\pi\cdot h\cdot{55\over64}={1375\over64}\cdot\pi\cdot h, egy petákért \left({1375\over64}\cdot\pi\cdot h\right):11={125\over64}\cdot\pi\cdot h köbcenti jár. Mivel {15\over8}={120\over64}<{125\over64}, ezért a Jókedvű Csordásban olcsóbb a kakaó.

Statistics on problem K. 55.
231 students sent a solution.
6 points:133 students.
5 points:17 students.
4 points:25 students.
3 points:11 students.
2 points:14 students.
1 point:8 students.
0 point:22 students.
Unfair, not evaluated:1 solution.

  • Problems in Mathematics of KöMaL, November 2005

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