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Work: The Pool Problem

  1. May 11, 2008 #1
    1. The problem statement, all variables and given/known data
    A circular swimming pool has a diameter of 22 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water over the side? (Use the fact that water weighs 62.5 lb/ft^3.)

    (a) Show how to approximate the required work by a Riemann sum. Give your answer using the form below. (If you need to enter -infinity or infinity, type -INFINITY or INFINITY.)

    (b)Express the work as an integral and evaluate it.

    3. The attempt at a solution
    I've found everything in the riemann sum equation minus the "E". What would this be?

    One I find that, I should be able to setup the integral and solve it.

  2. jcsd
  3. May 11, 2008 #2


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    Do you understand what that formula MEANS? Or where it came from. Frankly, to me it makes no sense. Unless there is an x "hidden" in the E it has the wrong units: the area of a circle is [itex]\pi r^2[/itex], not [itex]\pi r[/itex]!

    Imagine dividing the pool into layers- with each layer a specific depth x below the level of the water and thickness [itex]\Delta x[/itex]. That "layer" has area [itex]\pi(11^2)[/itex] square feet and so volume [itex]121\pi \Delta x[/itex] cubic feet. All the water in that "layer" has weight [itex](62.5)121\pi \Delta x[/itex] pounds and it must be raised x+ 5 feet (5 feet up to the water level and another 5 feet to the top of the pool). How much work is required to do that? The total work is found by summing over all the different "[itex]\Delta x[/itex] thickness layers. You turn that Riemann sum into an integral by taking [itex]\Delta x[/itex] smaller and smaller- becoming dx. The limits of integration will be from 0 (the top of the pond) to 4 (the bottom).
  4. May 11, 2008 #3


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    Can you come up with a mathematical expression describing the amount of work done [tex]\delta W[/tex] for each [tex]\delta V[/tex] infinitesimal volume of water to be raised from its current depth in the pool to its height?
  5. May 12, 2008 #4
    Thanks HallsOfIvy & Defennder. Well, now i've solved the problem and found the amount of work, but the elusive "E" variable in that question escapes me. :rofl:
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