If x2 + y2 + z2 = 9, dxdt = 4, and dydt = 8, find dzdt when (x, y, z) = (2, 2, 1).

To solve this problem, we need to use the chain rule of differentiation.

Given the equation x² + y² + z² = 9, we can differentiate both sides with respect to t to get:

2x(dx/dt) + 2y(dy/dt) + 2z(dz/dt) = 0.

We know that dx/dt = 4, dy/dt = 8, and we are trying to find dz/dt. We also know that at the point (x, y, z) = (2, 2, 1), the values of x, y, and z are 2, 2, and 1 respectively.

Substituting these values into the differentiated equation, we get:

2(2)(4) + 2(2)(8) + 2(1)(dz/dt) = 0.

Solving this equation for dz/dt, we get:

16 + 32 + 2(dz/dt) = 0, 48 + 2(dz/dt) = 0, 2(dz/dt) = -48, dz/dt = -48 / 2, dz/dt = -24.

So, when (x, y, z) = (2, 2, 1), dz/dt = -24.