Let C be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. Find the exact length of C from the origin to the point (3, 9/2, 9/2).

To find the exact length of the curve C from the origin to the point (3, 9/2, 9/2), we need to parametrize the curve C and then calculate the arc length.

Step 1: Parametrize the curve C.
The curve C is the intersection of the parabolic cylinder x^2 = 2y and the surface 3z = xy. We can substitute y = x^2/2 into the equation for z to get:
3z = x(x^2/2)
z = x^3/6
So the parametric equations for C are:
x = t
y = t^2/2
z = t^3/6

Step 2: Calculate the derivative of the parametric equations.
To calculate the derivative of the parametric equations, differentiate each equation with respect to t:
dx/dt = 1
dy/dt = t
dz/dt = t^2/2

Step 3: Set up the integral for arc length.
The arc length of the curve C from t = a to t = b is given by the integral:

L = ∫[(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2]^0.5 dt, where the limits of integration are from 0 to b.

Plugging in the derivatives from step 2, the integral becomes:

L = ∫[(1)^2 + (t)^2 + (t^2/2)^2]^0.5 dt

Simplifying the integral, we get:

L = ∫[(1 + t^2 + t^4/4)^0.5] dt, with limits of integration from 0 to b.

Step 4: Evaluate the integral.
Unfortunately, evaluating this integral analytically can be quite challenging and may not lead to a simple closed-form solution. In this case, you can rely on numerical methods or approximation techniques to find an approximate value for the length of the curve C.

There are several numerical methods you can use, such as numerical quadrature methods like Simpson's rule or the trapezoidal rule, or numerical integration using software programs like MATLAB or Python.

Remember to use the specific limits of integration based on the given endpoint (3, 9/2, 9/2) to obtain the exact length of the curve C.

To find the length of the curve C from the origin to the point (3, 9/2, 9/2), we need to parameterize the curve C and then integrate the magnitude of its derivative from the origin to the given point.

First, let's find the parameterization of the curve C. We can express y in terms of x using the equation of the parabolic cylinder:

x^2 = 2y
=> y = x^2/2

Now, let's substitute this expression for y into the equation of the surface to find z:

3z = xy
=> z = xy/3
=> z = x*(x^2/2)/3
=> z = x^3/6

So, the parameterization of the curve C is given by:
r(t) = (t, t^2/2, t^3/6)
where t represents the parameter along the curve.

To find the exact length of the curve C from the origin to the point (3, 9/2, 9/2), we need to evaluate the integral of the magnitude of the derivative of r(t) with respect to t, over the range of t from 0 to 3.

The magnitude of the derivative of r(t) is given by:
|dr/dt| = √( (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 )

Let's calculate the derivative of r(t) component-wise:
dx/dt = 1
dy/dt = t
dz/dt = t^2/2

Substituting these values, we get:
|dr/dt| = √(1 + t^2 + (t^2/2)^2)
= √(1 + t^2 + t^4/4)
= √(4 + 4t^2 + t^4)/2
= (2 + t^2)/2
= 1 + t^2/2

Now, we integrate |dr/dt| over the range of t from 0 to 3:
∫(0 to 3) (1 + t^2/2) dt = ∫(0 to 3) dt + ∫(0 to 3) (t^2/2) dt
= [t] from 0 to 3 + (1/2) ∫(0 to 3) t^2 dt
= 3 + (1/2) * [ (1/3) t^3 ] from 0 to 3
= 3 + (1/2) * (1/3) * 3^3 - (1/2) * (1/3) * 0^3
= 3 + (1/2) * (1/3) * 27
= 3 + 9/2
= 15/2

So, the exact length of the curve C from the origin to the point (3, 9/2, 9/2) is 15/2.

Therefore, the length of C is 15/2.