In my final board exam(not CBSE but Maharashtra board) we had a group D question which went like this:
If we put an inverted cone in a cylinder full of water, the cylinder being of the same radius and height, how much water will spill..(and the radius and heights were given.)Even then, my memory was like a bowl containing petrol kept in hot summer sun. Things just evaporated out of it. And I could not remember what the formula for the volume of a cone was. I knew it was similar to that of a cylinder but that was the most my hard disc could store.
I tried hard but could not reach any corner of my head where i could find even a slightest hint of the formula. Ironically, almost 10 years later, I still remember exactly, that i was trying enough to remember the day when my teacher (Dhond madam) taught us mensuration.So that if not the formula I could remember the day and then get some cues to figure out the formula. Of course I couldn't.
I knew then that I was to somehow discover the formula myself. As if it was possible. But I had then and have since in many exams made an honest attempt in discovering formulas in exams. The formulas for which scientists have got nobel prices and which could have been easily learnt 10 minutes before the exams. Of course, I am never successful with those.
But then my attempts always give me some formula or the other. I remember this one particularly because this was my first. I asked for a supplementary sheet for rough work. This itself was odd because other students had done every problem asked in the paper at least 3-4 times either as practice in school or tuition or mock test. Their efforts were on writing the steps left aligned and reasons right aligned separated by a crisp line drawn with an Apsara extra dark pencil, sharpened with a mental Natraj sharpener religiously after every question attempted. I remember being jealous of the guy sitting next to me.
Anyways, so on the rough paper I drew a rectangle standing on its shorter side. In it i drew an inverted triangle, with triangle's base overlapping the rectangle's top and its tip touching the rectangle's base.
I thought that if I take this paper and cut it along the rectangle. Then if i attach a slightly long matchstick along the altitude of the triangle with a portion of the stick protruding out of the base of the rectangle. The I hold this flag like structure with the stick between my palms and rub my palms, the rectangle will rotate along the altitude of the triangle and so will the triangle. Now if I do it fast enough, won't it resemble the given problem of a cone inside a cylinder! I was excited, I knew I had found it. Yes!!
Calming myself down i thought that because triangle's area is half of the rectangle's, the volume of the cone should be half the volume of the cylinder. Because after all i can now create a cone given thousand sheets of paper with a triangle drawn on them placed next to each other slightly twisted along the altitude. So i quickly inferred the formula that volume of a cone is (1/2)*pi*r^2*h.
I only remember that "Aha!" moment which I had on arriving at this wrong answer on my own, I don't remember the regret i would surely have had once i realized that i was wrong, after coming out of the board exam.