10. [-/1 Points] MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER If $x^{2}+y^{2}+z^{2}=9, \frac{d x}{d t}=7$, and $\frac{d y}{d t}=6$, find $\frac{d z}{d t}$ when $(x, y, z)=(2,2,1)$. \[ \frac{d z}{d t}= \] Need Help? Read It Submit Answer
Solution
Step 1 ：We are given the equation \(x^{2}+y^{2}+z^{2}=9\) and the rates of change of \(x\) and \(y\) with respect to \(t\). We are asked to find the rate of change of \(z\) with respect to \(t\) when \((x, y, z)=(2,2,1)\).
Step 2 ：We can use the chain rule to differentiate the given equation with respect to \(t\). The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
Step 3 ：In this case, we can differentiate both sides of the equation with respect to \(t\) to get \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} + 2z\frac{dz}{dt} = 0\).
Step 4 ：We can then substitute the given values and solve for \(\frac{dz}{dt}\).
Step 5 ：Substituting \(dx/dt = 7\), \(dy/dt = 6\), \(x = 2\), \(y = 2\), and \(z = 1\) into the equation, we get \(2*2*7 + 2*2*6 + 2*1*dz/dt = 0\).
Step 6 ：Solving for \(dz/dt\), we get \(dz/dt = -26\).
Step 7 ：Final Answer: The rate of change of \(z\) with respect to \(t\) when \((x, y, z)=(2,2,1)\) is \(\boxed{-26}\).
