Question

Physics with calculus. I have an inverted frustrum of a right circular cone. Top radius is 10ft and bottom radius is 4ft and height is 4ft. It is full of water Find the work required to pump the water out of a spout at the top of the tank. use 62.5lb/ft^3 for the weight of water and 3.14 for pi.

Answer 1

That would be equivalent to the work you'd need to do to raise all the water in the container to the height of the spout. You could, for instance, calculate the total gravitational potential energy of the system and then calculate the total potential of the system if all the water were at the height of the spout and subtract -- that would give you the amount of work you'd need to put in.

You could start like this:
\[ PE = \int \rho g h dV = \rho g \int h dV = \rho g \int h \pi r^2 dh = \pi\cdot \rho g \int h\cdot r^2(h) dh\]
Now just express the radius as a function of height and you can do the integral.