What is the answer?

Question

5. A spherical tank with a radius of 3 meters is filled with water. How much work is required to pump the water to a height 5 m above the top of the tank?

a. Pick a location for the origin.

b. Based on that location, label the location values of the bottom, center, and top of the tank and of the outlet.

c. Sketch a thin, representative slice of water and label its location and its thickness.

d. Find the volume of that slice.

e. Set up and solve the integral.

(𝜌 = 1000, 𝑔 = 9.8).

Answer 1

Set the origin at the bottom of the tank with the vertical axis passing from the bottom thru the center

Let x be distance from this axis to the side of the tank noting that it forms the radius of a circle

Let r be the radius of the tank thus r²=x²+(r-h)². Solve to yield for x²=2rh-h² .

Note that pix²=area of a circle at height h and ρgπx²dh=dw (w = weight of the differential element).

Substituting yields dw= ρgπ(2rh-h²)dh

Note that dw must be lifted 5+r-h meters thus dE=(5+r-h)dw

Integrate dE from h=0 to h=2r to find the energy required to empty the tank ignoring the small amount that remains in the pipe above the tank at h=2r and h=2r+5