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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 = 26?

(Round your answers to three decimal places. Enter your answers from smallest to largest z-value.)

  • Calculus -

    The sphere, S, is given by:
    S(x,y,z): x²+y²+z²-26=0

    The helix, r, is given by:
    r(x,y,z): sin(t),cos(t),t
    which means that the helix is constrained to the cylinder of unit radius, C: C(x,y): x²+y²=1

    Thus, the solution for the intersections is given by the solution of
    S(sin(t), cos(t), t)=0
    from which it can be deduced by inspection that t=5 gives an exact solution, since
    (sin²(5)+cos²(5)) + 5²
    =(1) + 25

    Similarly, t=-5 is a solution.

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