Problem K. 392. (November 2013)
K. 392. An underground supporting cable is to be fixed to a pole that will be entirely under ground, even the top of the pole will be buried. When the pit dug for the pole was half done, the workers lowered the pole into the pit. They made the following observation: When the whole pit is dug out, the top of the pole will be exactly twice as much below ground level as it is sticking out of the pit now. Given that the height of the pole is two metres, find the depth of the pit when it is completely dug out.
Deadline expired on December 10, 2013.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. Ha a teljesen kész gödör \(\displaystyle 2x\) méter mély lesz, akkor az oszlop teteje \(\displaystyle 2x-2\) méterrel lesz a föld alatt. Mikor félig kész a gödör, akkor \(\displaystyle x\) méter mély, és az oszlop teteje \(\displaystyle 2-x\) méterrel lóg ki belőle. Tudjuk, hogy \(\displaystyle 2\cdot(2-x)=2x-2\), ebből \(\displaystyle 4-2x=2x-2\), vagyis \(\displaystyle 6=4x\), és innen \(\displaystyle x=6/4=1,5\). A gödör \(\displaystyle 2\cdot x=2\cdot1,5=3\) méter mély lesz.
233 students sent a solution. 6 points: 154 students. 5 points: 32 students. 4 points: 16 students. 3 points: 5 students. 2 points: 13 students. 1 point: 11 students. Unfair, not evaluated: 2 solutionss.