Work required to pump water out of conical tank

[akbar786]

Homework Statement

Find the work done in pumping all the water out of a conical reservoir of radius 10ft at the top and altitude 8ft if at the beginning the reservoir is filled to a depth of 5ft and the water is pumped just to the top of the reservoir.

Homework Equations

None

The Attempt at a Solution

This is my work so far. I am taking the integral from -8 to -3. The top center of the cone is on (0,0). I have the integral going from -8 to -3 of (100 pi) *(62.4) (0-y) dy. The 100 is from the radius squared. The (0-y) is the distance the water has to travel up any given y. Any help?

 

[Dick]

You want to integrate pi*r^2*(-y)*dy from -8 to -3, right? So you want to express r as a function of y, don't you? Why are you using 100, the radius squared at the top and where did 62.4 come from?

 

[akbar786]

You want to integrate pi*r^2*(-y)*dy from -8 to -3, right? So you want to express r as a function of y, don't you? Why are you using 100, the radius squared at the top and where did 62.4 come from?

62.4 is the density of water the teacher wanted us to use.I decided to keep all my numbers positive which will also make it much easier to integrate.Here is my new integral with expressing r as a function of y. 62.4*pi * integral from 0 to 5 of ((5/4y)^2) * (8-y)
8-y is the distance the water has to travel given the generic y and on each of those y's the radius will be 5/4 of the y term. Is this right? I am solving for work and 62.4 is my density for water

 

[Dick]

62.4 is the density of water the teacher wanted us to use.I decided to keep all my numbers positive which will also make it much easier to integrate.Here is my new integral with expressing r as a function of y. 62.4*pi * integral from 0 to 5 of ((5/4y)^2) * (8-y)
8-y is the distance the water has to travel given the generic y and on each of those y's the radius will be 5/4 of the y term. Is this right? I am solving for work and 62.4 is my density for water

Now that looks right to me. Putting the origin at the bottom does make it much less confusing.

 

[akbar786]

Awesome, thanks a lot for your help.