A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5lb/ft3.
The work required to pump the water out of the spout.
The work is calculated by using the formula
Area of circular disk is
From the given figure observe that the given tank is spherical in shape.
The radius of the tank below the center is and from the tap to the center, the height is .Let centre be orgin and let the vertical down the centre and passing through the north and south pole of sphere be x-axis. The x-axis is positive downwards.
Consider circular disk slice of water having thickness lying at the coordinate . Its’ radius
Volume of the disk of water
Therefore, the weight of the slice is N
This weight is lifted by the pump to about height.s
Therefore, the work needed to pump the slice water out of the spout is
Therefore, the total work reqiured is
In the integral the function is odd since the degree of is which is an odd number, therefore, the value of
And in the integral the function is even since the degree of x is 2 which is even therefore,
The work required to pump the water out of the spout is J