Question

At what points does the helix r(t) = sin(t), cos(t), t intersect the sphere x² + y² + z² = 10?

Answer 1

The helix intersects the sphere at the points (sin(3), cos(3), 3) and (sin(-3), cos(-3), -3).

Step-by-step explanation:

The helix equation is given by r(t) = sin(t), cos(t), t, and the equation of the sphere is x² + y² + z² = 10. To find the points where the helix intersects the sphere, we can substitute the helix equation into the sphere equation.

Substituting the x, y, and z values of the helix equation into the sphere equation, we get sin²(t) + cos²(t) + t² = 10.

Simplifying this equation, we have 1 + t² = 10.

Therefore, t² = 9 and t = ±3. So the helix intersects the sphere at the points (sin(3), cos(3), 3) and (sin(-3), cos(-3), -3).