If z^2 = x^2 + y^2 with z>0, dx/dt=3, and dy/dt=6, find dz/dt when x=4 and y=3.
Answer: \displaystyle \frac{dz}{dt} =
2z dz/dt = 2x dx/dt + 2y dy/dt
when x=4 and y = 3
z^2 = 4^2 + 3^2 = 25
z = 5
Now that you have 5 of the 6 terms, just sub those in and solve for dz/dt
To solve this problem, we can use implicit differentiation and the chain rule.
Given z^2 = x^2 + y^2, we can differentiate both sides with respect to t:
2z * dz/dt = 2x * dx/dt + 2y * dy/dt
Now we can substitute the given values to solve for dz/dt. Given dx/dt = 3, dy/dt = 6, x = 4, and y = 3:
2z * dz/dt = 2x * dx/dt + 2y * dy/dt
2z * dz/dt = 2(4)(3) + 2(3)(6)
2z * dz/dt = 24 + 36
2z * dz/dt = 60
Now we can solve for dz/dt by dividing both sides by 2z:
dz/dt = 60 / (2z)
dz/dt = 30 / z
Since we are given that z > 0, we can substitute the given values of x = 4 and y = 3 to find the value of z:
z^2 = x^2 + y^2
z^2 = 4^2 + 3^2
z^2 = 16 + 9
z^2 = 25
z = 5
Now we can substitute the value of z into the equation for dz/dt:
dz/dt = 30 / z
dz/dt = 30 / 5
dz/dt = 6
Therefore, when x = 4 and y = 3, dz/dt = 6.