Question

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.)

Answer 1

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
=26

Similarly, t=-5 is a solution.