Arc Length & Derivative Calculations in Parametric Curves

A car travels along a path defined by the parametric equations x(t) = 3t^2 and y(t) = 2t^3 for t in [0, 2]. Calculate the arc length of the path from t = 0 to t = 2.

• 10

• 12

• 8

• 15

A particle moves along a curve defined by the parametric equations x(t) = cos(t) and y(t) = sin(t) for t in [0, 2π]. Determine the derivative of the parametric equations and find the velocity vector at t = π/4.

• (√2/2, -√2/2)

• (-1, 0)

• Velocity vector at t = π/4 is (-√2/2, √2/2).

• (0, 1)

A roller coaster's height is modeled by the parametric equations x(t) = 5t and y(t) = 10 - t^2 for t in [0, 3]. Calculate the arc length of the roller coaster's path from t = 0 to t = 3.

• L = 15.5

• L = 12.3

• L = 10.5

• L = 20.0

A drone follows a path described by the parametric equations x(t) = 4t and y(t) = 3t^2 for t in [0, 2]. Find the derivative of the parametric equations and interpret the result in terms of the drone's speed.

• The speed of the drone is 5t.

• The speed of the drone is 12t.

• The speed of the drone is sqrt(16 + 36t^2).

• The speed of the drone is 9t^2.

A boat moves in a circular path defined by the parametric equations x(t) = 5cos(t) and y(t) = 5sin(t) for t in [0, 2π]. Calculate the arc length of the boat's journey around the circle.

• 15π

• 10π

• 5π

• 20π

A cyclist rides along a path defined by the parametric equations x(t) = t^3 - 3t and y(t) = t^2 for t in [-2, 2]. Determine the derivative of the parametric equations and find the slope of the path at t = 1.

• undefined

• 2

• 1

• 0

A bird flies along a path described by the parametric equations x(t) = 2t and y(t) = 4 - t^2 for t in [0, 2]. Calculate the arc length of the bird's flight from t = 0 to t = 2.

• 3√5 - ln(1 + √5)

• 4 + 2√5

• 2√5 - ln(2 - √5)

• 2√5 + ln(2 + √5)

A runner's path is modeled by the parametric equations x(t) = 3t and y(t) = 2t^2 for t in [0, 4]. Find the derivative of the parametric equations and interpret the result in terms of the runner's acceleration.

• The runner accelerates horizontally at 2 units per time squared and has a vertical velocity of 3 units per time.

• The runner has a constant horizontal velocity and a vertical acceleration of 4 units per time squared.

• The runner has a vertical velocity of 4 units per time squared and a constant horizontal acceleration.

• The runner has a constant vertical velocity and a horizontal acceleration of 3 units per time squared.

A car's trajectory is defined by the parametric equations x(t) = 2t^2 and y(t) = 5t for t in [0, 3]. Calculate the arc length of the car's path from t = 0 to t = 3.

• L = 30.2

• L = 18.7

• L = 21.5

• L = 15.0

A skateboarder moves along a path defined by the parametric equations x(t) = t^2 and y(t) = 4 - t for t in [0, 4]. Determine the derivative of the parametric equations and find the instantaneous rate of change of y with respect to x at t = 2.

• -1

• 1.25

• -0.25

• 0.5