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Setting up the Integral

shamieh

Active member
Sep 13, 2013
539
All I need to do for this problem is set up the integral...Can someone tell me how to do that?

A tank has the shape of an inverted circular cone with height 10m and base with radius 1m. The tank is filled with water to a height of 8 m . Find the work required to empty the tank by pumping all of the water over the top.


NOTE: I just need to set up the integral, I don't actually have to calculate the problem.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
All I need to do for this problem is set up the integral...Can someone tell me how to do that?

A tank has the shape of an inverted circular cone with height 10m and base with radius 1m. The tank is filled with water to a height of 8 m . Find the work required to empty the tank by pumping all of the water over the top.


NOTE: I just need to set up the integral, I don't actually have to calculate the problem.
From the side the cone looks like a triangle and we can use similar triangles. With height h, the distance from the center of the cone to the side, r, we have r/h= 1/10 so that r= h/10. The area of the disk at that height is [tex]\pi r^2= \pi h^2/100[/tex] and the volume of a thin 'layer of water', with thickness dh is [tex]\frac{\pi}{100}h^2 dh[/tex]. Taking [tex]\delta[/tex] to be the density of water, it's weight is [tex]\frac{\pi\delta}{100}h^2dh[/tex]. Lifting that from height h to height 10m requires [tex]\frac{\pi\delta}{100}h^2(10- h)dh[/tex] Joules of work. Integrate that from h= 0 to h= 8.