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Volume of water in a pond HELP

  1. Jun 11, 2008 #1
    Volume of water in a pond HELP!!

    1) A pond is approximately circular, with a diameter of 400 feet. Starting at the center, the depth of the water is measured every 25 feet and recorded in the table below.

    Feet
    from
    Center 0 25 50 75 100 125 150 175 200
    Depth
    in
    Feet
    20 19 19 17 15 14 10 6 0


    a) One way to view the trapezoidal approximation of an integral is to say that on each subinterval you approximate by a first-degree polynomial. In Simpson’s Rule, named after the English mathematician Thomas Simpson (1710-1761), you take this procedure one step further and approximate
    by second-degree polynomials. Using Simpson’s Rule, estimate the volume of water in the pond. Try your best to ensure that your estimation, if in error, has an error that is less than 0.01!


    b) Determine a quadratic model that describes the relationship between the distance from center and the depth of the water. Please use the distance from center as your independent variable in this model.
    c) Provide an alternative strategy for estimating the volume of water in the pond, preferably one that involves familiar integration techniques and involves algebraic or symbolic manipulation.

    Now use the result of this integration to approximate the number of gallons of water in the pond.


    My attempt at an answer:

    Simpson's Rule:

    the integral from a to b of f(x) dx is approximately:
    ((b-a)/3n)[f(x(sub 0))+4f(x(sub 1))+2f(x(sub 2))+4f(x(sub 3))+...+4f(x(sub n-1))+f(x(sub n))]

    Using the rule I came up with 1,366,592.804 cubic feet for Part A but I am not sure that this is right...... as far as Parts B and C I am lost...

    Any help at all would be great!
     
    Last edited: Jun 11, 2008
  2. jcsd
  3. Jun 11, 2008 #2

    Tom Mattson

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    Staff Emeritus
    Science Advisor
    Gold Member

    We can't check your work without seeing what you did. How did you arrive at that number?
     
  4. Jun 11, 2008 #3
    Here's my work.....

    ((2pi(200))/(3(8)))[0+4(25)(19)+2(50)(19)+4(75)(17)+2(100)(15)+4(125)(14)+2(150)(10)+4(175)(6)+0]

    = (50pi/3)[26100]

    = 1,366,592.804 cubic feet
     
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