Calculus II: Work Problem - Find Work in Pumping Water to Top of Tank

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    Calculus ii Work
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SUMMARY

The problem involves calculating the work required to pump water from an inverted right-circular cone-shaped tank that is 29 ft across and 15 ft deep, with the water surface 5 ft below the top. The work is determined by integrating the weight of the water slices, which is the volume of each slice multiplied by the weight density W lbs. per cubic foot, over the height of the water. The formula for work is expressed as the integral of h multiplied by the volume of the water slice and W, covering the entire height of the water in the tank.

PREREQUISITES
  • Understanding of calculus concepts, specifically integration.
  • Familiarity with the geometry of cones and volume calculations.
  • Knowledge of weight density and its application in physics.
  • Basic principles of work in physics, defined as force times distance.
NEXT STEPS
  • Study integration techniques in calculus, focusing on applications in physics.
  • Learn about the geometry of solids, specifically the properties of cones.
  • Explore the concept of weight density and its calculations in fluid mechanics.
  • Review examples of work problems in physics, particularly those involving variable forces.
USEFUL FOR

Students studying calculus, particularly those focusing on applications in physics, as well as educators seeking to explain work problems involving fluid dynamics and integration techniques.

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"Work" (Calculus II)

This is a problem from the chapter called "Work":

A water tank in the form of an inverted right-circular cone is 29 ft across the top and 15 ft deep. If the surface of the water is 5 ft. below the top tank, find the work done in pumping the water to the top of the tank. Assume water weights W lbs. per cubic feet (which means your answer will be expressed in terms of W)


I have read it many times but I don't know what to do to solve it...please help, I really appreciate it.
 
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Work is weight times distance. At a given distance h below the top of the tank take a slice of the water of thickness dh. The weight is volume of the slice times W. The work to pump it to the top is h*(volume of the slice)*W. Now you need to integrate that over the range of h that includes all of the water in the tank. Is that enough to get started? It's a lot like finding the volume but with those extra factors of W and h.
 


thank you very much...i'm going to see if I can solve it using this :)
 

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