1. **State the problem:** A particle moves on the hyperbola defined by the equation $$xy = 15$$ for time $$t \geq 0$$ seconds. At a certain instant, $$x = 3$$ and $$\frac{dx}{dt} = 6$$. We need to find the rate of change of $$y$$ at this instant and determine whether $$y$$ is increasing or decreasing. 2. **Write the given equation and differentiate:** The hyperbola equation is $$xy = 15$$. Differentiate both sides with respect to time $$t$$ using implicit differentiation: $$\frac{d}{dt}(xy) = \frac{d}{dt}(15)$$ Using the product rule: $$x \frac{dy}{dt} + y \frac{dx}{dt} = 0$$ 3. **Solve for $$\frac{dy}{dt}$$:** $$x \frac{dy}{dt} = - y \frac{dx}{dt}$$ $$\frac{dy}{dt} = - \frac{y}{x} \frac{dx}{dt}$$ 4. **Calculate $$y$$ at $$x=3$$:** From the original equation: $$y = \frac{15}{x} = \frac{15}{3} = 5$$ 5. **Substitute known values:** $$\frac{dy}{dt} = - \frac{5}{3} \times 6 = -10$$ 6. **Interpret the result:** Since $$\frac{dy}{dt} = -10$$, $$y$$ is decreasing at a rate of 10 units per second at the instant when $$x=3$$. **Final answer:** A y is decreasing by 10 units per second.