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Math Help - simpsons rule application

  1. #1
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    simpsons rule application

    Using Simpsons Rule, answer the following question:

    1)
    The cross-sectional area of a fin is described by the area enclosed by the curves: y = x √(4-x), and the x-axis

    Each unit of the function is equivalent to 10 cm of actual length. If the fin is made from rigid Kevlar Fibre 1 cm thick, and Kevlar fibre costs $3.00 per square unit. can she keep her material costs for each fin below $25.00

    My attempt:

    Simpsons Rule: A ≈ ((b-a) / 3n) [ yo + 4*[sum of odd y's] + 2*[sum of even y's] + yf]

    y = x √(4-x)

    x-intercepts are 0 and 4, therefore b = 4, and a = 0

    Using n = 4, my intervals are each (4-0/4) units long or 1 unit


    y(0) = 0

    y(1) = 1.73
    y(2) = 2.82
    y(3) = 3

    y(4) = 0

    Area
    ≈ 1/3 [ 0 + 4(1.73 + 3) + 2(2.82) + 0]

    Area ≈ 8.18 square units, and since each unit = 10cm, the cost is equal to: $3.00 * (10*8.18) = $245.4
    This is above $25.00, therefore she will not meet the budget.

    What have I done wrong?
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  2. #2
    MHF Contributor
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    It really is interesting how often a student believe there is something horribly wrong, but no hint is given why the student feels this way. Please provide sufficient information for the discussion.

    In this case, if you have done anything "wrong", it would be failing to notice that you consistently rounded down.

    \sqrt{3}\;=\;1.732051... and you used 1.73, having lopped off 0.002051

    2 \cdot \sqrt{2}\;=\;2.828427... and you used 2.82, having lopped off 0.008427

    The other values are exact.

    In any case, you have grossly underestimated.

    Also, why did you multiply by 10? Each unit represents 10. Don't multiply by it again.

    3*(128/15) = 25.6 -- The exact value.

    3*(8.19502) = 24.585 -- Your value, but with more decimal places

    Isn't that interesting that we crossed 25, depending on how it is done? A very dangerous place to be.
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