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Math Help - please help me to check my answer and work.

  1. #1
    Member
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    Wink please help me to check my answer and work.

    Could you please help me to check my work for the below question? Thank you very much.

    ===============================
    Question. (calculus - work done)

    Find the work done by force field F on a particle that moves along the curve C
    F(x,y,z) = (x+y)i +xyj-z^2k
    C: along line segment from (0,0,0,) to (1,3,1) to (2,-1,4).

    Solution.
    let O=(0,0,0,), P=(1,3,1) and Q=(2,-1,4)

    For line segment OP,
    r(t)=<t, 3t,t> where 0< or = t < or = 1
    r'(t) = <1,3,1>

    F(r(t)) = <4t, 3t^2, -t^2>

    F(r(t)) dot r'(t) = 4t +8t^2

    there work of line segment OP = the integral from 0 to 1 of F(r(t)) dot r'(t) dt = 14/3

    For line segment PQ
    r(t)=<1+t, 3-4t, 1+3t> where 0< or = t < or = 1
    r'(t) = <1,-4, 3>

    F(r(t)) = <4-3t, 3-t-4t^2. -1-6t-9t^2>

    F(r(t)) dot r'(t) = -11-17t-11t^2


    there work of line segment PQ = the integral from 0 to 1 of F(r(t)) dot r'(t) dt = -139/6

    therefore the total work done of C : 14/3 + (-139/6) = -37/2

    ========================================
    Um... my work of this question is 2 pages.( that is too long )
    If you have better and faster approach to this question, please teach me. Thank you very much.
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  2. #2
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by kittycat View Post
    Could you please help me to check my work for the below question? Thank you very much.

    ===============================
    Question. (calculus - work done)

    Find the work done by force field F on a particle that moves along the curve C
    F(x,y,z) = (x+y)i +xyj-z^2k
    C: along line segment from (0,0,0,) to (1,3,1) to (2,-1,4).

    Solution.
    let O=(0,0,0,), P=(1,3,1) and Q=(2,-1,4)

    For line segment OP,
    r(t)=<t, 3t,t> where 0< or = t < or = 1
    r'(t) = <1,3,1>

    F(r(t)) = <4t, 3t^2, -t^2>

    F(r(t)) dot r'(t) = 4t +8t^2

    there work of line segment OP = the integral from 0 to 1 of F(r(t)) dot r'(t) dt = 14/3

    For line segment PQ
    r(t)=<1+t, 3-4t, 1+3t> where 0< or = t < or = 1
    r'(t) = <1,-4, 3>

    F(r(t)) = <4-3t, 3-t-4t^2. -1-6t-9t^2>

    F(r(t)) dot r'(t) = -11-17t-11t^2


    there work of line segment PQ = the integral from 0 to 1 of F(r(t)) dot r'(t) dt = -139/6

    therefore the total work done of C : 14/3 + (-139/6) = -37/2

    ========================================
    Um... my work of this question is 2 pages.( that is too long )
    If you have better and faster approach to this question, please teach me. Thank you very much.
    two pages? really? this whole solution would occupy only 3/4 of a page (or maybe less) or maybe it just depends on the font size of our penmanship..
    Good job!!
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