As the Crow Fills

Geometry Level 3

Once, a very thirsty crow searched for water everywhere. At last he saw a pot with but very little water in it. He put his beak inside. Alas, he could not reach down to the water level. The shrewd crow then thought of a plan. He picked some pebbles and dropped them one by one into the pot. The water level started rising till it came right up to the brim. The crow drank the water and congratulated itself for being clever. We’ve always known this story since we were little children, haven’t we? Now for the mathematics of the situation: Suppose the pot was fashioned with a hemisphere d= 40 cm and a truncated cone (base circle diameter of 40 cm and top circle diameter of 14 cm with a height of 14 cm). If the crow selected perfect spherical pebbles of 1 cm diameter i.e. quite small as compared to the pot size for filling the pot, what should have been the minimum level of water (in cms correct to 3 decimal places) initially in the spheri-conical pot if the crow did not go thirsty?

Problem from Mindsport by Mukul Sharma
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