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At what points does the helix r = sin(t), cos(t), t intersect the sphere x 2 + y 2 + z 2 = 37? (Round your answers to three decimal places. Enter your answers from smallest to largest z-value.)

  • calculus -

    A helix r=(sin(t),cos(t),t) is a coil like a spring of radius 1 around the z-axis.
    The coordinates in cartesian coordinates are:

    The distance from origin is
    When it intersects a sphere of radius R, the following relationship applies:
    sin²(t)+cos²(t)+t² = 37² ...(1)

    Using sin²(x)+cos²(x)=1, (1) simplifies to
    1+t²=37² ...(2)

    Solve 2 for t and back substitute to find x(t), y(t) and z(t)

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