Question
A spherical tank with radius of 5 feet is set on a coloumn 15 feet above the ground. How much work is required to fill the tank with water if the solution is pumped from ground level?
Answers
Steve
Let the center of the tank be at (0,0)
when the water level is at y, the radius of the water surface is
r^2 = 25-y^2
Work is ∫F(y) dy
F(y) is the weight of the water. So, since each slice of water in the tank is raised 21+y feet, and water weighs 62.4 lbs/ft^3,
W(y) = ∫[-5,5] 62.4 π (25-y^2) (21+y) dy
= 218400π
when the water level is at y, the radius of the water surface is
r^2 = 25-y^2
Work is ∫F(y) dy
F(y) is the weight of the water. So, since each slice of water in the tank is raised 21+y feet, and water weighs 62.4 lbs/ft^3,
W(y) = ∫[-5,5] 62.4 π (25-y^2) (21+y) dy
= 218400π