Let *C* be the curve of intersection of the parabolic cylinder *x*^2 = 2*y*, and the surface 3*z* = *xy*. Find the exact length of *C* from the origin to the point (5, 25/2 ,125/6). do not explain, just give answer

The exact length of curve *C* from the origin to the point (5, 25/2, 125/6) is:

\[
L = \frac{125}{6}
\]

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Here are 5 related questions to expand on this concept:
1. How is the length of a curve in 3D space calculated using parametric equations?
2. What is the geometric interpretation of the intersection of a parabolic cylinder and a surface?
3. How do you parametrize a curve given by two surfaces?
4. Can you calculate the length of a curve using polar coordinates?
5. How does the arc length formula differ in 2D and 3D?

**Tip:** When calculating the length of a curve, always double-check the parametric equations and their derivatives to ensure accuracy.

Let *C* be the curve of intersection of the parabolic cylinder *x*^2 = 2*y*, and the surface 3*z* = *xy*. Find the exact length of *C* from the origin to the point (5, 25/2 ,125/6).

Find the exact length of the curve *C*, formed by the intersection of the parabolic cylinder x^2 = 2*y and the surface 3*z = *xy*, from the origin to the point (5, 25/2, 125/6). This problem covers key concepts from multivariable calculus, including parametrization and the arc length formula.

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