Question

The distance between a surface and point is the minimum distance between the given point and any point on the surface. This distance can be found using constrained optimization. Find the minimum distance between the point P(5,−12,−39) and the cone: z²=x²+y² a. Find the distance between any point (x,y,z) and the point (5,−12,−39). Distance: b. Minimize the distance function subject to the constraint that the point (x,y,z) is on the above cone. Hint: Use Lagrange multipliers and since the square root is an increasing function it is easier to minimize the square distance. Minimum Distance:

Answer 1

The distance between a point and the cone is first expressed using the Euclidean distance formula. Then, to find the minimum distance, the distance function is minimized with the constraint of the cone's equation using Lagrange multipliers. The square distance, rather than the distance itself, is minimized for simplification.

Step-by-step explanation:

The distance between any point (x,y,z) and the point P(5,−12,−39) can be found using the Euclidean distance formula:

Distance = √[(x-5)2 + (y+12)2 + (z+39)2]

To minimize the distance function subject to the constraint that the point (x,y,z) lies on the cone z2=x2+y2, we use Lagrange multipliers. Because calculating the square root is not necessary when minimizing (as the square root function is monotonically increasing), we minimize the square of the distance instead:

D2 = (x-5)2 + (y+12)2 + (z+39)2

Applying the Lagrange multiplier technique, we set up the system:

• ∇D2 = λ∇g, where g(x,y,z) = x2 + y2 - z2 is the constraint function representing the cone.
• This yields a set of equations that can be solved for x, y, z, and λ.
• Substitute x, y, and z back into the distance formula to find the minimum distance.