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

In summary, the problem involves finding the work done in pumping water from a right-circular cone tank to the top, given the dimensions and weight of the water. The solution requires integrating the weight and distance of each slice of water over the entire range of the tank's depth.
  • #1
paiway
5
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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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  • #2


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.
 
  • #3


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

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

1. What is the purpose of solving work problems in Calculus II?

The purpose of solving work problems in Calculus II is to use mathematical techniques to find the amount of work done in a specific situation. This helps in understanding the relationships between force, distance, and work and can be applied to various real-life scenarios.

2. How is work calculated in the context of pumping water to the top of a tank?

The work done in pumping water to the top of a tank is calculated by multiplying the force applied (in this case, the force from the pump) by the distance over which the force is applied (the height of the tank). This can be represented as W = F * d.

3. What is the role of integration in solving work problems?

Integration is used in solving work problems to find the total work done by continuously summing up the small amounts of work done at each point along the path. This is represented by the definite integral in the formula W = ∫F(x)dx, where F(x) is the force function and dx represents the infinitesimal distance.

4. How does the shape of the tank affect the work calculation?

The shape of the tank does not affect the work calculation as long as the height and force remain constant. This is because the work done only depends on the vertical distance the water is pumped, not the shape of the tank.

5. Can Calculus II be used to solve work problems in other scenarios?

Yes, Calculus II can be applied to solve work problems in various scenarios, such as lifting objects, moving fluids, and compressing gases. The same principles of force, distance, and work can be used to calculate the total work done in these situations.

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