Final answer:
The shortest distance from the surface xy+3x+z^2=9 to the origin can be found using the distance formula. To find the point on the surface that is closest to the origin, we need to minimize the distance formula by finding the point where the gradient of the surface is perpendicular to the line connecting the point to the origin. Once we find this point, we can use the distance formula to calculate the shortest distance.
Step-by-step explanation:
The shortest distance from the surface xy+3x+z2=9 to the origin can be found using the distance formula. The distance from a point (x,y,z) to the origin (0,0,0) is given by the formula D = sqrt(x^2 + y^2 + z^2). In this case, we want to find the shortest distance, so we need to minimize this formula.
To minimize D, we need to find the point on the surface that is closest to the origin. This can be done by finding the point where the gradient of the surface is perpendicular to the line connecting the point to the origin.
The gradient of the surface is given by the vector (∂F/∂x, ∂F/∂y, ∂F/∂z), where F(x,y,z) = xy+3x+z2-9. To find the shortest distance, we set the dot product of this gradient vector with the vector (x, y, z) (which represents the line connecting the point to the origin) equal to 0.
Simplifying the dot product equation, we have (∂F/∂x)x + (∂F/∂y)y + (∂F/∂z)z = 0. Substituting the partial derivatives of F, we get (y+3)x + xy + 2z = 0.
Now we have a system of equations (the equation of the surface xy+3x+z2=9 and the dot product equation) that we can solve to find the values of x, y, and z that correspond to the point on the surface that is closest to the origin. Once we know these values, we can plug them into the distance formula D = sqrt(x^2 + y^2 + z^2) to find the shortest distance.