Digging a Well (Work Problem)

• Sarangalex
In summary: It is just a matter of which end you put your coordinate system and whether x is measured positive up or down.No, I was thinking of a wedge-shaped excavation (because of the word "prism" in the original post). In that case I would be wrong anyway, because (my) dW is weight, not work. For a wedge-shaped hole the answer would beWork = int rho * 40 * x*(1 - x/30) dx = 900000 ft-lb.

Homework Statement

A well is dug in the shape of a rectangular prism. It is 30ft deep and has a base with area of 40ft^2. Assuming that the soil weighs 150 lbs/ft^3, calculate the work W required to raise the soil to ground level.

Homework Equations

W = ∫dW
dW = ρ(area)(distance)dx

The Attempt at a Solution

Then I said W = 150∫[from 0 to 30] 1200-40x dx

Solved to get 2700000ft/lb, which is wrong. I don't really understand what I did wrong here. Isn't this just like pumping water out of a tank or something?

I get the same answer. I guess we are both assuming "rectangular prism" means the same thing as "rectangular parallelepiped" or "rectangular box".

Sarangalex said:

Homework Statement

A well is dug in the shape of a rectangular prism. It is 30ft deep and has a base with area of 40ft^2. Assuming that the soil weighs 150 lbs/ft^3, calculate the work W required to raise the soil to ground level.

Homework Equations

W = ∫dW
dW = ρ(area)(distance)dx

The Attempt at a Solution

Then I said W = 150∫[from 0 to 30] 1200-40x dx

Solved to get 2700000ft/lb, which is wrong. I don't really understand what I did wrong here. Isn't this just like pumping water out of a tank or something?

I think your answer is too large by a factor of 30. Go back to your dW formula to see why.

RGV

I don't understand why it would just be x for the distance rather than 30-x. Could someone explain this to me?

It doesn't matter whether you use x or 30 - x. I think you are correct and I am puzzled what RGV sees that we don't, since his answers are usually spot on.

But x and 30-x give you totally different answers...

Sarangalex said:
But x and 30-x give you totally different answers...

No they don't.

LCKurtz said:
It doesn't matter whether you use x or 30 - x. I think you are correct and I am puzzled what RGV sees that we don't, since his answers are usually spot on.

The cross-sectional area at depth x is 40 at x = 0, but the formula given by the OP gives 40*30. I think the correct formula should be dW = ρ*40*(30-x)/30 dx.

RGV

Ray Vickson said:
The cross-sectional area at depth x is 40 at x = 0, but the formula given by the OP gives 40*30. I think the correct formula should be dW = ρ*40*(30-x)/30 dx.

RGV

But the OP has just chosen his coordinates measuring depth from the other end; that's the 30-x, which varies from 30 to 0. You could measure from the other end and use x as I mentioned earlier. The cross section is a constant 40.

We are both thinking of a hole shaped by a rectangular box, right?

Oh, I'm sorry. I see they are the same now, I just don't understand how.

Sarangalex said:
Oh, I'm sorry. I see they are the same now, I just don't understand how.

It is just a matter of which end you put your coordinate system and whether x is measured
positive up or down.

No, I was thinking of a wedge-shaped excavation (because of the word "prism" in the original post). In that case I would be wrong anyway, because (my) dW is weight, not work. For a wedge-shaped hole the answer would be
Work = int rho * 40 * x*(1 - x/30) dx = 900000 ft-lb.

RGV

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1. How deep should a well be dug?

The depth of a well depends on several factors such as the water table level, soil type, and intended use of the well. In general, most residential wells are dug to a depth of 100-400 feet.

2. What equipment is needed to dig a well?

The equipment needed to dig a well includes a drilling rig, drill bits, mud pumps, and casing. The specific equipment may vary depending on the type and depth of the well.

3. How long does it take to dig a well?

The time it takes to dig a well can vary greatly depending on the depth, type of soil, and equipment used. On average, it can take anywhere from a few days to a few weeks to dig a well.

4. Is it safe to drink water from a well?

Yes, it is generally safe to drink water from a well. However, it is important to have the water tested regularly to ensure it is free of harmful bacteria and contaminants. It is also recommended to install a filtration system for added safety.

5. Can a well run dry?

Yes, it is possible for a well to run dry if the water table level drops significantly. This can happen due to drought, overuse of the well, or changes in the surrounding environment. Regular maintenance and monitoring can help prevent a well from running dry.