Consider the curve C parametrized by
x = cos (8t) and y = sin (8t), for
0 ≤ t ≤ 𝜋/8
and traversed by a particle moving along C with increasing values of t. Choose the correct answer below.
The particle travels clockwise, traversing 1/16 of the unit circle.
The particle travels counterclockwise, traversing 1/16 of the unit circle.
The particle travels clockwise, traversing 1/8 of the unit circle .
The particle travels counterclockwise, traversing 1/8 of the unit circle.
The particle travels clockwise, traversing 1/2 of the unit circle.
The particle travels counterclockwise, traversing 1/2 of the unit circle.
The particle travels clockwise, 8 times around the unit circle.
The particle travels counterclockwise, 8 times around the unit circle.
Incorrect: Your answer is incorrect.
no it isnt
as t goes from 0 to π/8, 8t goes from 0 to π
so, counterclockwise halfway around the circle
To determine the direction and distance the particle travels along the curve, we can analyze the behavior of the parametric equations x = cos(8t) and y = sin(8t).
Since the x-coordinate of a point on the unit circle oscillates between -1 and 1 as t increases, we can conclude that the parametric equation x = cos(8t) traces a portion of the unit circle.
Similarly, since the y-coordinate of a point on the unit circle oscillates between -1 and 1 as t increases, we can conclude that the parametric equation y = sin(8t) also traces a portion of the unit circle.
The parameter t ranges from 0 to 𝜋/8, which means the particle travels for 1/8th of a complete revolution since 𝜋/8 is 1/8th the value of 𝜋 (a complete revolution).
Now, let's determine the direction of travel. Since the x-coordinate (cos(8t)) is positive and decreasing from 1 to -1 as t increases, we can conclude that the particle travels clockwise along the unit circle.
Therefore, the correct answer is:
"The particle travels clockwise, traversing 1/8 of the unit circle."