Question

Let \((a, b, c)\) be a fixed point in the first octant. Find the plane through this point that cuts off from the first octant the tetrahedron of minimum volume, and determine the resulting volume.

Short answer

The minimum volume is \( \frac{4abc}{3} \).

Step-by-step solution

Understand the Problem

We are tasked with finding a plane that intersects the coordinate axes at points \(x_0, 0, 0\), \(0, y_0, 0\), and \(0, 0, z_0\), and passes through the fixed point \(a, b, c\)\(\). This plane will cut off a tetrahedron in the first octant, and we want to find the plane that minimizes the volume of this tetrahedron.

Equation of the Plane

The general equation of a plane in three-dimensional space is \(rac{x}{x_0} + rac{y}{y_0} + rac{z}{z_0} = 1\). Since this plane passes through \(a, b, c\), the coordinates must satisfy the plane equation, resulting in \(rac{a}{x_0} + rac{b}{y_0} + rac{c}{z_0} = 1\).

Volume of the Tetrahedron

The volume \(V\) of a tetrahedron with vertices at the origin \( (0,0,0) \), \( (x_0,0,0) \), \( (0,y_0,0) \), and \( (0,0,z_0) \) is given by \(rac{x_0 y_0 z_0}{6}\). We want to minimize this volume given the constraint from the plane equation.

Set Up the Optimization Problem

We want to minimize the function \(V = \frac{x_0 y_0 z_0}{6}\) subject to the constraint \(\frac{a}{x_0} + \frac{b}{y_0} + \frac{c}{z_0} = 1\). We can use the method of Lagrange multipliers to find the minimum volume condition.

Use Lagrange Multipliers

Introduce a Lagrange multiplier \( ext{ }\lambda\) and set up the Lagrangian \(L(x_0, y_0, z_0, \lambda) = \frac{x_0 y_0 z_0}{6} + \lambda \igg(\frac{a}{x_0} + \frac{b}{y_0} + \frac{c}{z_0} - 1\bigg)\), and solve the system of equations derived from \(rac{\partial L}{\partial x_0} = 0 \), \(rac{\partial L}{\partial y_0} = 0\), \(rac{\partial L}{\partial z_0} = 0\), and \(rac{\partial L}{\partial \lambda} = 0\).

Solve the System to Find Critical Points

Differentiate the Lagrangian and set equations: \( y_0z_0 = \lambda (\frac{a}{x_0^2}) \, x_0z_0 = \lambda (\frac{b}{y_0^2}) \, x_0y_0 = \lambda (\frac{c}{z_0^2}) \, and \frac{a}{x_0} + rac{b}{y_0} + rac{c}{z_0} = 1\). Solving these, we find \( x_0 = 2a\), \( y_0 = 2b\), \( z_0 = 2c\).

Calculate the Minimum Volume

Substitute \(x_0 = 2a\), \(y_0 = 2b\), and \(z_0 = 2c\) back into the volume function: \( V = rac{(2a)(2b)(2c)}{6} = rac{8abc}{6} = rac{4abc}{3}\).

Verify and Conclude

Verify the solution satisfies all requirements and conclude that the plane through \(a, b, c\) cuts off the tetrahedron with the minimum volume of \(\frac{4abc}{3}\) from the first octant.

Key concepts

Three-Dimensional Geometry

Three-dimensional geometry is all about understanding shapes and figures that exist in space. Imagine how a box or a ball looks different from a square or a circle. Three-dimensional geometry involves length, width, and height.

For this problem, the plane that intersects the coordinate axes forms a three-dimensional shape called a tetrahedron. A tetrahedron is a pyramid-like figure with four triangular faces, one on each side. When solving problems in three-dimensions, it's essential to understand how volumes and positions of shapes work in space.

• Axes are like rulers measured from zero in different directions.
• Points are described with x, y, and z coordinates, like addresses in space.
• A plane is a flat surface that extends forever in three dimensions.

By using these elements, you can describe and solve spatial problems like finding where a plane cuts the axes and forms a minimum-sized tetrahedron.

Lagrange Multipliers

The concept of Lagrange multipliers is a smart mathematical tool used in optimization problems. Optimization involves finding maximums or minimums—like the smallest volume of a tetrahedron.

To understand Lagrange multipliers, picture having a path on a map but needing to pass through a particular point for a minimum journey. It's about controlling the path while adhering to a set constraint.

• Function:
  The main function in this exercise is the volume of the tetrahedron.
• Constraint:
  The constraint is the plane's equation derived from its location in space.
• Lambda (𝜆):
  A clever trick in the math that helps you find the optimum solution by balancing function and constraint.

Lagrange multipliers give a systematic way to find out how to adjust the variables (coordinates of the tetrahedron) to get the minimum volume while maintaining the given spatial conditions.

Volume of a Tetrahedron

The volume of a tetrahedron is the space contained within its four triangular faces. Calculating the volume of such shapes is a key part of geometry. The vertices of this tetrahedron are at the origin and along the x, y, and z axes.

For this problem, the important volume formula is given as:
\[ V = \frac{x_0 y_0 z_0}{6} \]

• Origin:
  The starting point is at coordinates (0,0,0).
• Vertices:
  Located at the points where the plane meets the axes.

This expression helps to compute how much space the tetrahedron occupies. A tetrahedron's volume can be influenced by its dimensions. Thus, minimizing the volume, while keeping the fixed point in place, involves adjusting the coordinates x, y, and z according to given constraints.

Coordinate Geometry

Coordinate geometry is where algebra meets geometry. It enables us to describe geometric figures like lines, planes, and solids using numbers. In this problem, we use coordinate geometry to express the positioning of objects in three-dimensional space.

The equation of a plane in coordinate geometry looks like:
\[ \frac{x}{x_0} + \frac{y}{y_0} + \frac{z}{z_0} = 1 \]
This equation is significant because it defines the location of our plane in space. It also lets us determine where the plane intersects the x, y, and z axes, forming a tetrahedron.

• x, y, z coordinates:
  Specifies a point's position in space.
• Intersection points:
  These are special points where the plane crosses the axes.

Utilizing these concepts, you gain insights into how the plane's equation ties into locating the axes' intersections to solve for minimizing the tetrahedron's volume.