A hemispherical tank, 10 ft in diameter is full of liquid (density: 40 lb/ft3). Find the work required to pump the water out of the top of the tank.

The work required to pump a liquid to a certain height is given by the formula:

W = ρghA

where:

• W is the work done,
• ρ is the density of the liquid,
• g is the acceleration due to gravity,
• h is the height to which the liquid is pumped,
• A is the area of the liquid surface.

In this case, we are given that the density of the liquid is 40 lb/ft³, the acceleration due to gravity is approximately 32.2 ft/s², and the tank is a hemisphere with a diameter of 10 ft, so the radius is 5 ft.

The work done to pump the liquid out of the tank is the sum of the work done to pump each infinitesimally small slice of liquid at height y to the top of the tank. This is given by the integral from y=0 to y=5 of ρgπ(5^2 - y^2)(5-y) dy, where (5^2 - y^2) is the area of the slice at height y and (5-y) is the distance the slice needs to be pumped.

So, the work done is:

W = ∫ from 0 to 5 of 4032.2π*(5^2 - y^2)*(5-y) dy

This integral can be solved using standard calculus techniques. The result is the work required to pump the liquid out of the tank.