A particle moves along the curve 6y = x³ + 2. The point on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate is:

A particle moves along the curve 6y = x³ + 2. The point on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate is:

• A. (4, 11)
• B. (2, 5/3)
• C. (−2, −1)
• D. (6, 109/3)

Answer: A) (4, 11)

Explanation: 6y = x³ + 2 → 6 dy/dt = 3x² dx/dt. Given dy/dt = 8 dx/dt → 6(8 dx/dt) = 3x² dx/dt → 48 = 3x² → x² = 16 → x = ±4. For x = 4, 6y = 64 + 2 = 66 → y = 11. Point (4, 11). For x = −4, y = −62/6 = −31/3, not in options.