Question

At what points does the helix r(t) = sin(t), cos(t), t intersect the sphere x² + y² + z² = 65?

Answer 1

The helix intersects the sphere at the points (sin(8), cos(8), 8) and (sin(-8), cos(-8), -8).

Step-by-step explanation:

The helix can be represented by the parametric equations:

x = sin(t)

y = cos(t)

z = t

The equation of the sphere is given by:

x^2 + y^2 + z^2 = 65

Substituting the parametric equations into the equation of the sphere, we get:

sin^2(t) + cos^2(t) + t^2 = 65

Since sin^2(t) + cos^2(t) = 1, we can simplify the equation to:

t^2 + 1 = 65

t^2 = 64

t = ±8

Therefore, the helix intersects the sphere at the points (sin(8), cos(8), 8) and (sin(-8), cos(-8), -8).