Question

Let r(t) = < sin(6t), cos(6t), sin(6t)cos(12t) >.

Find the point where r(t) intersects the xy-plane on the interval π/6 < t < 3/12π.

Answer 1

When r(t) intersects the xy-plane, z = 0
in the interval π/6 < t < 3π/12 or π/6 < t < π/4 , (think 30° < t < 45°)
so sin(6t)cos(12t)= 0
so sin 6t = 0 or sin 12t = 0

if sin 6t = 0 , then 6t = 0 , π, 2π, ..
t = 0, π/6 , π/3, π/2 , ... (0°, 30° 60°, ..)
since we want π/6 < t < π/4, none of the answers falls within that domain.

if sin 12t = 0, then 12t = 0, π, 2π, 3π, 4π, 5π, 6π, ...
t = 0, π/12, π/6, π/4, π/3 .... (0, 15°, 30°, 45°, 60°....

again, since your domain is π/6 < t < π/4, none of your answers satisfy.

(There are several places were it would touch the xy-plane, had it been
π/6 ≤ t ≤ π/4 )