Final answer:
The shortest distance from the surface xy+6x+z²=41 to the origin is approximately 4.57 units.
Step-by-step explanation:
To find the shortest distance from the surface xy+6x+z²=41 to the origin, we need to find the perpendicular distance from the origin to the surface. The perpendicular distance is equal to the absolute value of the constant term in the equation divided by the square root of the coefficients of x, y, and z squared. In this case, the distance is √(41 / (1² + 0² + 1²)) = √(41 / 2) ≈ 4.57 units.