Analyzing Motion and Intersections in Parametric Equations

A car's motion is described by the parametric equations x(t) = 3t and y(t) = 2t^2. Determine the position of the car at t = 4 seconds and analyze its motion.

• (15, 40)

• (8, 16)

• (10, 20)

• The position of the car at t = 4 seconds is (12, 32).

A particle moves along a path defined by the parametric equations x(t) = 5cos(t) and y(t) = 5sin(t). Find the coordinates of the particle at t = π/4 and describe its motion.

• (5, 5)

• (5√3/2, 5/2)

• (5√2/2, 5√2/2)

• (0, 0)

Two friends are walking along paths defined by the parametric equations A: x(t) = 2t + 1, y(t) = t^2 and B: x(t) = t^2, y(t) = 2t. Find the points where their paths intersect.

• (0, 0)

• (3, 4)

• (1, 2)

• [ (2 + 2√2, 3 + 2√2), (2 - 2√2, 3 - 2√2) ]

A drone follows a path described by the parametric equations x(t) = 4t and y(t) = 3t - 2. Calculate the position of the drone at t = 3 seconds and analyze its trajectory.

• (15, 9)

• (10, 5)

• The position of the drone at t = 3 seconds is (12, 7).

• (12, 5)

A ball is thrown with a path defined by the parametric equations x(t) = 10t and y(t) = -5t^2 + 20t. Determine the time when the ball reaches its maximum height and analyze its motion.

• 3 seconds

• 1 second

• 4 seconds

• 2 seconds

A cyclist travels along a circular path defined by the parametric equations x(t) = 10cos(t) and y(t) = 10sin(t). Find the coordinates of the cyclist at t = π/2 and describe the motion.

• (0, -10)

• (-10, 0)

• (10, 0)

• (0, 10)

Two curves are defined by the parametric equations C1: x(t) = t^2 and y(t) = 3t and C2: x(t) = 4t - 1 and y(t) = 2t + 1. Find the points of intersection of these curves.

• (12 + 6√6, 9 + 3√6) and (12 - 6√6, 9 - 3√6)

• (8 + 4√2, 6 + 2√2)

• (10, 6)

• (14 - 3√3, 12 + 3√3)

A boat moves in a river described by the parametric equations x(t) = 2t and y(t) = 5 - t^2. Calculate the position of the boat at t = 2 seconds and analyze its motion.

• (4, 4)

• (6, 0)

• The position of the boat at t = 2 seconds is (4, 1).

• (2, 1)

A roller coaster's path is modeled by the parametric equations x(t) = 5t and y(t) = -4t^2 + 20t. Determine the time when the roller coaster reaches the ground and analyze its trajectory.

• 5 seconds

• 10 seconds

• 7 seconds

• 3 seconds

A soccer ball is kicked with a path defined by the parametric equations x(t) = 15t and y(t) = -4.9t^2 + 10t. Find the time when the ball hits the ground and analyze its motion.

• 1.5 seconds

• Approximately 2.04 seconds

• 4.2 seconds

• 3.0 seconds