The value of c, where c is the ratio of the radius of the water surface to the height of the cone is √2π/√3 m.
Here since the vertex is downwards, the water filled up to the depth of 2 m takes a shape of a cone with a height of 2 m.
Let the radius of that cone be r m.
The rate of the increase in water level is 5 m per minute at a 2m level.
Hence the time is taken to fill 2 m up
= 2/5 min
It is given that the speed of water pouring is 10 m³/min. Therefore, the volume of the water-filled
= 2/5 X 10 m³
= 4 m³
The volume of a cone = 1/3 πr²h
where r = radius of the cone and h = height.
putting the value of volume and height we get
4 = 1/3 πr² X 2
or, πr² = 2 X 3
or, r² = 6/π
or, r = √(6/π) m
Therefore, the ratio of the radius of the water surface by height will be
√(6/π) / 2
= 1/ (2/√(6/π))
According to the problem, this ratio is equal to 1: c
Hence c = 2/√(6/π)
= √2π/√3 m
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