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.
Well, to find where r(t) intersects the xy-plane, we need to find the value of t where the z-coordinate of r(t) is equal to zero. Let's put on our investigation hats and get to it!
The z-coordinate of r(t) is sin(6t)cos(12t), and we want to find when it equals to zero. Now, clowns may not be great at math, but we know one thing for sure: sin(0) is equal to zero. So, we can set sin(6t) equal to zero and solve for t.
sin(6t) = 0
By using basic trigonometry, we know that sin(θ) = 0 when θ is a multiple of π. So, 6t must be a multiple of π.
6t = πn
But, we also have the interval constraint: π/6 < t < 3/12π. Let's check if any multiples of π for 6t fall within this interval.
For π/6 < t, we have:
6t > π/6
t > π/36
Similarly, for t < 3/12π:
6t < 3/12π
t < π/8
Looking at both inequalities, we see that the values of t that satisfy both conditions are:
π/36 < t < π/8
Now, we have our interval where r(t) intersects the xy-plane. Now let's enjoy a little dance on the xy-plane together, shall we?
To find the point where r(t) intersects the xy-plane, we need to set the z-coordinate of r(t) equal to zero.
Given that r(t) = < sin(6t), cos(6t), sin(6t)cos(12t) >, the z-coordinate is sin(6t)cos(12t).
Setting sin(6t)cos(12t) = 0, we have two possible cases:
Case 1: sin(6t) = 0
In this case, sin(6t) = 0 implies that 6t = nπ for some integer n.
Hence, t = nπ/6.
Case 2: cos(12t) = 0
In this case, cos(12t) = 0 implies that 12t = (2n + 1)π/2 for some integer n.
Hence, t = (2n + 1)π/24.
Since we want the intersection point on the interval π/6 < t < 3/12π, we need to find the values of t that satisfy this condition.
For sin(6t) = 0, the values of t that satisfy the condition are t = π/6 and t = 5π/6.
For cos(12t) = 0, the values of t that satisfy the condition are t = (2n + 1)π/24 for n = 0, 1, 2, ..., 11.
Since we are given the interval π/6 < t < 3/12π, we need to find the values of n that satisfy this condition.
For n = 0, 2, 4, 6, 8, 10, the corresponding values of t are:
t = π/24, 13π/24, 25π/24, 37π/24, 49π/24, 61π/24.
Therefore, the points where r(t) intersects the xy-plane on the interval π/6 < t < 3/12π are:
(π/6, 0, 0)
(5π/6, 0, 0)
(π/24, 13π/24, 0)
(13π/24, 25π/24, 0)
(25π/24, 37π/24, 0)
(37π/24, 49π/24, 0)
(49π/24, 61π/24, 0)
To find the point where the vector r(t) intersects the xy-plane, we need to find the value of t that satisfies the condition z = 0.
The given vector r(t) can be written as r(t) = ⟨sin(6t), cos(6t), sin(6t)cos(12t)⟩.
To find z = 0, we need to solve the equation sin(6t)cos(12t) = 0.
To do this, consider the two factors separately:
1) sin(6t) = 0:
Setting sin(6t) = 0, we get 6t = nπ, where n is an integer.
Therefore, t = nπ/6.
2) cos(12t) = 0:
Setting cos(12t) = 0, we get 12t = (2n+1)(π/2), where n is an integer.
Therefore, t = (2n+1)(π/24).
Now, we need to find the intersection point on the interval π/6 < t < 3/12π.
Substituting the values of t obtained from the first factor sin(6t) = 0, we get t = {π/6, π/3, π/2, 2π/3, 5π/6}.
Substituting the values of t obtained from the second factor cos(12t) = 0, we get t = {π/24, 5π/24, 9π/24, 13π/24, 17π/24}.
Now, identifying the values of t that lie in the given interval, we see that π/6 < t < 3/12π only includes t = {π/3, π/2, 2π/3}.
Substituting these values of t back into the equation r(t) = ⟨sin(6t), cos(6t), sin(6t)cos(12t)⟩, we find the intersection points on the xy-plane:
r(π/3) = ⟨sin(6(π/3)), cos(6(π/3)), sin(6(π/3))cos(12(π/3))⟩
= ⟨0, 1/2, 0⟩
r(π/2) = ⟨sin(6(π/2)), cos(6(π/2)), sin(6(π/2))cos(12(π/2))⟩
= ⟨1, 0, 0⟩
r(2π/3) = ⟨sin(6(2π/3)), cos(6(2π/3)), sin(6(2π/3))cos(12(2π/3))⟩
= ⟨0, -1/2, 0⟩
Hence, the points where r(t) intersects the xy-plane on the interval π/6 < t < 3/12π are (0, 1/2, 0), (1, 0, 0), and (0, -1/2, 0).
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 )