Water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base.
Given: The water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m.
To find how fast the water level is rising when the water stands 7.5 m below the base.
Let the height of the cone be H = 10m (given)
Let the radius of the base be R = 5m (given)
Let O’Y = r and CO’ = h
Now from the above figure,
ΔCOB~ΔCO’Q
So,
Let the volume of the water in the vessel at any time t be V
Then,
(from equation (i))
Now differentiate the above equation with respect to t, we get
But given the water is running at the rate of m3/min, i.e.,
So the above equation becomes
So when the water stands 7.5 m below the base
So h = 10 - 7.5 = 2.5m, the rate becomes
Hence the rate of water level rising when the water stands 7.5m below the base is 0.64 metres per min