A particle is constrained to move on the surface x42x3zyz0

A particle is constrained to move on the surface x4+2x3z?yz=0 . At the point (1, 3, 1), y is changing at the rate of 2m/sec and z is changing at the rate of 1m/sec. Determine the rate of change(with respect to time)of ?z?x at (1, 3, 1).

Solution

x⁴+2x³z-yz =0

on differentiating wrt time

4x³ dx/dt + 6x²z dx/dt + 2x³ dz/dt - y dz/dt - zdy/dt =0

putting every values given

4dx/dt + 6 dx/dt + 2 x 1 - 3x 1 -1 x2 =0 at (1,3,1)

=> dx/dt = 3/10 = 0.3m/sec

therefore rate of change in x is 0.3 m/sec.