A clown is juggling at a circus. The path of a ball is given by the parametric equations x=2 cos T + 2 and y = 3 sin T + 3. In what direction is the ball moving?
Up and to the right
counterclockwise
down and to the right
clockwise
After graphing it and observing its transformation, it seems to be clockwise but I just want to be sure and have someone check my work. If it's wrong, can someone please explain to me why the correct answer would be correct.
no!!
it is an ellipse. when t=0 you are at the right side, ready to move up and to the left: counterclockwise.
http://www.wolframalpha.com/input/?i=plot+x%3D2cost%2B2,+y%3D3sint%2B3,+t%3D0+to+pi%2F2
better review how to plot points!
well,
x starts at a max and decreases
y starts at the center point and increases.
if that doesn't help, note that the ball is moving
from (4,3) toward (2,6)
care to try again?
would it be Down and to the right?
Thank you Steve!
To determine the direction in which the ball is moving, we need to analyze the parametric equations of its path.
The given parametric equations are:
x = 2cos(T) + 2
y = 3sin(T) + 3
By examining these equations, we can see that x and y are functions of T, which represents time or a parameter.
To understand the direction of the ball's movement, we can consider the derivatives of x and y with respect to T. The derivatives represent the rate of change of the coordinates x and y with respect to time.
By using the chain rule and differentiating both equations with respect to T, we obtain:
dx/dT = -2sin(T)
dy/dT = 3cos(T)
Now, let's analyze the derivatives individually.
dx/dT represents the rate of change of the x-coordinate with respect to time. In this case, dx/dT = -2sin(T).
Since sin(T) has a maximum value of 1 at T = π/2 and a minimum value of -1 at T = 3π/2, we can conclude that -2sin(T) will be at its maximum (magnitude) when T = π/2 and at its minimum (magnitude) when T = 3π/2.
Therefore, dx/dT will have a maximum negative value when T = π/2 (180 degrees) and a minimum negative value when T = 3π/2 (270 degrees). This means that the x-coordinate is decreasing as T increases from 0 to 2π.
Similarly, dy/dT represents the rate of change of the y-coordinate with respect to time. In this case, dy/dT = 3cos(T).
Since cos(T) has a maximum value of 1 at T = 0 and a minimum value of -1 at T = π, we can conclude that 3cos(T) will be at its maximum (magnitude) when T = 0 and at its minimum (magnitude) when T = π.
Therefore, dy/dT will have a maximum positive value when T = 0 (0 degrees) and a minimum negative value when T = π (180 degrees). This means that the y-coordinate is increasing as T increases from 0 to π and then decreasing as T increases from π to 2π.
Putting it all together:
- When dx/dT is negative and dy/dT is positive, the ball is moving down and to the right. This is not the case here.
- When dx/dT is negative and dy/dT is negative, the ball is moving up and to the left. This is not the case here.
- When dx/dT is positive and dy/dT is positive, the ball is moving up and to the right. This is the case here.
- When dx/dT is positive and dy/dT is negative, the ball is moving down and to the left. This is not the case here.
Therefore, based on the derivatives dx/dT and dy/dT, the ball is moving up and to the right.