Calculus Work Problem (Spherical Water Tower)

[dbai]

Homework Statement

A spherical water tower 40 ft in diameter has its center 120 ft above the ground. That means, there is a 120 ft pole connected to the 40 ft diameter spherical tank. Water is being pumped at the ground level to fill the tank with water of density 62.4 lb/ft³,

a) How much work is done in filling the tank half full?
b) How much work is done in filling the tank ENTIRELY full?

Homework Equations

dW = (dV)*(density)*(displacement)

The Attempt at a Solution

• I calculated that dV would be a cylindrical section of the tank with height dy --> πx² dy
• Density is given 62.4 lb/ft³
• I used (120 - y) as my displacement
  
  ∫(Pi)x²(120 - y) dy Limits: 0-120 ??
  
  I don't know if the displacement is correct and how to convert x in terms of y.

 

[Saladsamurai]

I am not sure what it is are using x for? Do you mean r?

 

[Defennder]

What is x supposed to be here? You should draw a picture of a right-angle triangle whose L-shape rests upright on the infinitesimal cylindrical slice of water. Then the hypotenuse would be constant, namely 20^2. Denote the base by x (or r) and the height as 20-y. See how to take to it from here?