To solve this problem, let’s break down the approach that has been used.

Problem Statement

We need to find the minimum distance from the point $(3, 4, 15)$ to the surface defined by $x^2 + y^2 - 4z^2 = 0$.

Solution Outline

1. Distance Formula: The distance $D$ from any point $(x, y, z)$ on the surface to the point $(3, 4, 15)$ is given by: $D = \sqrt{(x - 3)^2 + (y - 4)^2 + (z - 15)^2}$ However, for simplification, we work with the square of the distance function: $f(x, y, z) = (x - 3)^2 + (y - 4)^2 + (z - 15)^2$

2. Constraint: The point $(x, y, z)$ lies on the surface defined by $\phi(x, y, z) = x^2 + y^2 - 4z^2 = 0$.

3. Lagrange Multipliers: To minimize $f(x, y, z)$ subject to the constraint $\phi(x, y, z) = 0$, we set up the Lagrange function: $F(x, y, z, \lambda) = f(x, y, z) + \lambda \phi(x, y, z)$ Expanding it: $F(x, y, z, \lambda) = (x - 3)^2 + (y - 4)^2 + (z - 15)^2 + \lambda (x^2 + y^2 - 4z^2)$

4. Partial Derivatives: To find the minimum, we take partial derivatives with respect to $x$, $y$, $z$, and $\lambda$, and set them equal to zero.

   • For $\frac{\partial F}{\partial x} = 0$: $2(x - 3) + 2\lambda x = 0 \Rightarrow -\lambda = \frac{x - 3}{x} \quad \text{(Equation 1)}$

   • For $\frac{\partial F}{\partial y} = 0$: $2(y - 4) + 2\lambda y = 0 \Rightarrow -\lambda = \frac{y - 4}{y} \quad \text{(Equation 2)}$

   • For $\frac{\partial F}{\partial z} = 0$: $2(z - 15) - 8\lambda z = 0 \Rightarrow -\lambda = \frac{z - 15}{4z} \quad \text{(Equation 3)}$

5. Solving the System: Using Equations (1), (2), and (3), we set up relationships between $x$, $y$, and $z$ in terms of $\lambda$ and solve for the minimum point satisfying both the constraint and the Lagrange conditions.

Would you like a step-by-step solution on solving these equations, or is there a specific part of the derivation you'd like further clarification on?

Related Questions:

1. How do Lagrange multipliers help in constrained optimization problems?
2. What is the geometric interpretation of the constraint $x^2 + y^2 - 4z^2 = 0$?
3. Can we solve this problem without using Lagrange multipliers? If so, how?
4. What are some common applications of distance minimization in real-world problems?
5. How does changing the coordinates of the fixed point affect the minimum distance?

Tip

When using Lagrange multipliers, double-check each partial derivative for accuracy.