Final answer:
To find dz/dt at (x,y)=(1,3) and z²=x²+y², we first need to find dz/dx and dz/dy. From the given equation z²=x²+y², we can differentiate both sides with respect to x. Then, we can use the chain rule to find dz/dt.
Step-by-step explanation:
To find dz/dt at (x,y)=(1,3) and z²=x²+y², we first need to find dz/dx and dz/dy. From the given equation z²=x²+y², we can differentiate both sides with respect to x. Taking the derivative of z² with respect to x gives us 2z dz/dx = 2x. Solving for dz/dx, we get dz/dx = x/z. Similarly, differentiating both sides of the given equation with respect to y gives us 2z dz/dy = 2y. Solving for dz/dy, we get dz/dy = y/z.
Now, to find dz/dt, we can use the chain rule. dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt). Substituting the given values dx/dt=4, dy/dt=3, x=1, y=3, and z=√(x²+y²)=√(1²+3²)=√10, we get dz/dt = (1/√10)(4) + (3/√10)(3) = 4/√10 + 9/√10 = 13/√10 = 13√10/10 ≈ 4.11.