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?

Question

(1 point)
Responses

up and to the right
up and to the right

counterclockwise
counterclockwise

down and to the right
down and to the right

clockwise

Answer 1

To determine the direction in which the ball is moving, we will analyze the given parametric equations for the path of the ball:

• \( x(t) = 2 \cos t + 2 \)
• \( y(t) = 3 \sin t + 3 \)

These equations represent an ellipse centered at the point (2, 3) with a semi-major axis of 3 (vertical direction) and a semi-minor axis of 2 (horizontal direction).

To determine the direction of motion, we can compute the derivatives of the parametric equations with respect to time \( t \):

1. For \( x(t) \): \[ \frac{dx}{dt} = -2 \sin t \]

2. For \( y(t) \): \[ \frac{dy}{dt} = 3 \cos t \]

The direction of the motion can be inferred from the signs of these derivatives.

• When \( t = 0 \):
  • \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 3 \) (moving upwards)
• When \( t = \frac{\pi}{2} \):
  • \( \frac{dx}{dt} = -2 \) and \( \frac{dy}{dt} = 0 \) (moving left)
• When \( t = \pi \):
  • \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = -3 \) (moving downwards)
• When \( t = \frac{3\pi}{2} \):
  • \( \frac{dx}{dt} = 2 \) and \( \frac{dy}{dt} = 0 \) (moving right)

Combining these movements shows that the motion follows the path of an ellipse in a counterclockwise direction, as it progresses from upward to left, downward, and then to the right.

Therefore, the ball is moving in a counterclockwise direction.

The correct response is: counterclockwise