A particle moves on the hyperbola xy=18 for time t≥0 seconds. At a certain instant, y=6 and dydt=8. Which of the following is true about x in this instant?

Answer 1

x = 3 and , differentiating implicitly with respect to t , ydx/dt + xdy/dt = 0 so 6dx/dt +24 = 0 and dx/dt = -4.

This rate of change for x makes sense in 2 ways: since the hyperbola is monotone decreasing, it follows that when dy/dt is + , dx/dt must be - . Also, solving for y explicitly gives y = 18/x. Then differentiating with respect to x this time gives us dy/dx = - 18 / x². At the point (3,6) , dy/dx = - 2 which is, gladly, = dy/dt / dx/dt.