Final answer:
The length of the curve of intersection between the surfaces is calculated using parameterization and integration of the arc length formula. The final result for the length of the curve from the point (0,0,0) to the point (6, 18, 36) is option c) 12√5.
Step-by-step explanation:
To find the length of the curve of intersection between the parabolic cylinder x² = 2y and the plane 3z = xy from the point (0,0,0) to the point (6, 18, 36), we can parameterize the curve using one of the variables and then integrate the arc length differential.
Let's use x as our parameter. Since x² = 2y, we have y = x²/2. From the second equation, 3z = xy, and substituting for y, we get z = x³/6.
Now we find the derivatives of y with respect to x, which is dy/dx = x, and the derivative of z with respect to x, which is dz/dx = x²/2.
The formula for the arc length, L, from a to b is given by ∫ sqrt((dx/dx)² + (dy/dx)² + (dz/dx)²) dx, which is ∫ sqrt(1 + x² + x´/4) dx from 0 to 6.
The integral simplifies to ∫ sqrt(1 + x² + x´/4) dx, and after evaluating, we find that the length is 12√5, which is option c) 12√5.