Calculus

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An open hemispherical bowl of radius R is slowly
draining through a small hole of radius r in the bottom
of the bowl. Let g be the acceleration due to gravity
and y be the depth of the water (which means the distance
from the water surface to the hole). Torricelli’s
Law states that the speed of the droplets leaving the hole is
√2gy . (This is actually the
speed that the droplets would acquire by falling from the water surface to the hole.)
(a) What is the meaning of the expression πr2√2gy ?
(b) In terms of y, what is the area of the water surface when the water is y cm deep?
(c) Assume that the bowl is initially full. How much time will be needed to drain it?

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