Final answer:
The helix r(t) = (sin(t), cos(t), t) intersects the sphere x² + y² + z² = 10 at two points, which are determined by substituting the parametric equations into the sphere's equation and solving for t, resulting in the points (sin(3), cos(3), 3) and (sin(-3), cos(-3), -3).
Step-by-step explanation:
To determine at what points the helix r(t) = (sin(t), cos(t), t) intersects the sphere x² + y² + z² = 10, we must substitute the parametric equations of the helix into the equation of the sphere. The helix provides us with the values of x, y, and z in terms of the parameter t: x = sin(t), y = cos(t), and z = t. Substituting these into the sphere's equation gives us sin(t)² + cos(t)² + t² = 10.
Since sin(t)² + cos(t)² is always equal to 1, the equation simplifies to 1 + t² = 10, or t² = 9. The solutions for t are t = ±3. Hence, the helix intersects the sphere at two points, which correspond to t = 3 and t = -3.
Substituting t = 3 into the helix equations gives us the point (sin(3), cos(3), 3), and substituting t = -3 gives us the point (sin(-3), cos(-3), -3). Therefore, these are the points of intersection between the helix and the sphere.