Judith B.

asked • 07/15/14

Work pumping oil out of a tank

A spherical tank is half full of oil that has a density of 900 kg/m^3. Find the work required to pump the oil out of the spout on top. Use 9.8 for g and 3.14 for pi. Round to 3 sig. figs. 
The radius of the sphere is 10.5 m. The height of the spout is 3.5 m.
 
i know that the density times volume is mass. The mass times 9.8 will give me the force. I assumed I needed to get the volume of a "slice" of the oil and figure out how much work it takes to pump it out. Then integrate to sum up the work for all of the oil to be removed. But I'm clearly doing something wrong and I am uncertain if the distance is to the top of the sphere or the top of the spout. 
 
Can someone set me straight?

Philip P.

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Judith - I have posted my understanding of the geometry of your problem here:
 
http://www.wyzant.com/resources/files/277867/work_to_pump_oil_from_a_spherical_tank
 
Please look it over and see if correctly models your problem.  I have set up the integral but you must complete it.
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07/16/14

Judith B.

That's awesome Phillip. Your diagram really helps. it didn't occur to me to use a triangle in this instance. I was using the equation of a circle and trying to relate height and radius through that. Now I'm cooking with gas :)
Thanks a ton. I definitely appreciate your teaching style.
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07/16/14

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Philip P. answered • 07/15/14

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You are right in that you will need to perform an integration of incremental slices of the oil.  The solution is a bit involved and requires some diagramming which is beyond the feeble capability of the Wyzant text editor.  So instead, I'm sending you to the website below:
 
http://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx
 
Example #4 on that web page is a very similar problem to yours but with a different shaped tank.  The solution process is the same, however.
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Judith B.

Thank you Philip for responding. I've seen that problem worked. I'm not sure how to apply it. I know I need the volume of an ith slice but how do I use the radius of the sphere to get the volume of a generic slice? I'm not sure what relationship to use. And when I do the "work" to get it to the top, should it be the entire distance up the sphere plus the spout? 
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07/15/14

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I have already solved the complete problem.
The work done comes out to be negative.
This is because of wrong spout height.
If you like I can send you the complete solution.
So, finally the question is wrong.
 
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07/16/14

Inactive Tutor answered • 07/15/14

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The tank is already half full of water, up to a height of 10.5 mt.
The spout is at a height of 3.5 mt.
So the water will come out by itself without doing any work, till the height of water in the tank is 3.5 Mt.
 
Surendra
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Judith B.

No, that's not the case. The spout is on top of the sphere.
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07/15/14

Inactive Tutor

If the spout is on the top, how its height can be 3.5 Mt???????
If this height is above the level of water then it is ok.
But that is NOT mentioned in the question.
 
 
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07/15/14

Judith B.

The spout measures 3.5 m tall. I can't diagram it for you, but in the problem it shows it as h for the height of the spout. So the top of the spout is at total height 24.5 m. As I said in the original question, the spout is on top and the height of the spout is 3.5. Remember, I'm the student. I asked for Help. I may not have been clear enough on the question, but you needn't berate or belittle me. 
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07/15/14

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