Work required to pump water out of conical tank

In summary, the water is pumped up to the top of the reservoir. The integral is solved to find the work done. The radius and density are used to find the work.
  • #1
akbar786
18
0

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?
 
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  • #2
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?
 
  • #3
Dick said:
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
 
  • #4
akbar786 said:
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.
 
  • #5
Awesome, thanks a lot for your help.
 

1. How is the work required to pump water out of a conical tank calculated?

The work required to pump water out of a conical tank is calculated by multiplying the weight of the water by the height it needs to be lifted and the gravitational constant (9.8 m/s²). This can be represented by the formula W = mgh, where W is the work required, m is the mass of the water, g is the gravitational constant, and h is the height the water needs to be pumped.

2. Does the shape of the conical tank affect the work required?

Yes, the shape of the conical tank does affect the work required to pump water out of it. The conical shape means that the volume of water decreases as it is pumped out, resulting in a decrease in weight and therefore less work required to lift it.

3. How does the diameter of the tank's base impact the work required?

The diameter of the tank's base does not directly impact the work required to pump water out of it. However, a wider base may result in a larger volume of water, which would require more work to pump out. Additionally, a wider base may also mean a longer distance for the water to be pumped, which would also require more work.

4. Can the work required be reduced by using a more powerful pump?

Yes, using a more powerful pump can reduce the amount of work required to pump water out of a conical tank. A more powerful pump can lift the water to a greater height in a shorter amount of time, reducing the overall work required. However, using a more powerful pump may also increase the cost and energy consumption.

5. Are there any other factors that can impact the work required to pump water out of a conical tank?

Yes, there are several other factors that can impact the work required to pump water out of a conical tank. These include the density of the water (which can vary depending on temperature and impurities), the efficiency of the pump, and any friction or resistance in the pumping system. Additionally, the elevation of the tank and the air pressure can also affect the work required.

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