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Math Help - Parametric Arc Length and Tangent Equations

  1. #1
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    Parametric Arc Length and Tangent Equations

    Find the exact length of the curve:

    x=5+15t^2
    y=6+2t^3
    0, greater than or = to, t, greater than or = to, 4

    The other problem is:

    Find equations of tangents to x=3t^2+8, y=2t^3+8, that passes through (11,0).

    I had NO idea how to do those. I did a couple steps of each and got to something that I'm unsure of. Any help would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by thegame189 View Post
    Find the exact length of the curve:

    x=5+15t^2
    y=6+2t^3
    0, greater than or = to, t, greater than or = to, 4

    The other problem is:

    Find equations of tangents to x=3t^2+8, y=2t^3+8, that passes through (11,0).

    I had NO idea how to do those. I did a couple steps of each and got to something that I'm unsure of. Any help would be greatly appreciated!
    The Formula for arc length is s=\int_{t_0}^{t_1}\sqrt{\left(\frac{dx}{dt}\right)  +\left(\frac{dy}{dt}\right)}dt=\int_0^4\sqrt{(30t)  ^2+(6t)^2}dt=6\sqrt{26}\int_0^4tdt=

    3\sqrt{26}t^2|_0^4=48\sqrt{26}

    For the second...

    \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=  \frac{6t}{6t}=1

    So the slope of the function is m=1. You should be able to finish from here.
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  3. #3
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    The first problem is y=6+2t^3, not squared.
    You end up with (30t)^2 + (6t^2)^2 under the radical. How do I do that?
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  4. #4
    Behold, the power of SARDINES!
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    Whoops

    Okay lets try again.

    <br />
s=\int_{t_0}^{t_1}\sqrt{\left(\frac{dx}{dt}\right)  +\left(\frac{dy}{dt}\right)}dt=\int_0^4\sqrt{(30t)  ^2+(6t^2)^2}dt=6\int_0^4t\sqrt{25+t^2}dt=<br />

    so let u=25+t^2 and du=2tdt

    so we get...

    3\int_{25}^{41}\sqrt{u}du=2u^{3/2}|_{25}^{41}=2(41\sqrt{41}-125)
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