At what points does the helix r = sin(t), cos(t), t intersect the sphere x 2 + y 2 + z 2 = 37? (Round your answers to three decimal places. Enter your answers from smallest to largest z-value.)

Answer 1

A helix r=(sin(t),cos(t),t) is a coil like a spring of radius 1 around the z-axis.
The coordinates in cartesian coordinates are:
x(t)=sin(t)
y(t)=cos(t)
z(t)=t

The distance from origin is
r=√(sin²(t)+cos²(t)+t²)
When it intersects a sphere of radius R, the following relationship applies:
sin²(t)+cos²(t)+t² = 37² ...(1)

Using sin²(x)+cos²(x)=1, (1) simplifies to
1+t²=37² ...(2)

Solve 2 for t and back substitute to find x(t), y(t) and z(t)