Math Problem Statement

The linear inequalities, for which the shaded area in the following figure is the solution set, are:

Solution

To determine the set of linear inequalities that correspond to the shaded region in the figure, let's analyze the lines and the region they bound. The figure seems to represent a triangular region formed by three planes in the first octant.

Analyzing the Planes

  1. Identify the equations of the planes:

    • The line intersecting the XX-axis at 4, the YY-axis at 2, and the ZZ-axis at 1 can be represented as: \frac{x}{4} + \frac{y}{2} + z = 1 \quad \text{(or equivalently, x + 2y + 4z = 4)}
    • The second line passes through the YY-axis at 2 and the ZZ-axis at 2, which can be represented as: \frac{y}{2} + \frac{z}{2} = 1 \quad \text{(or y + z = 2)}
    • The third line passes through the XX-axis at 8 and the ZZ-axis at 2: \frac{x}{8} + \frac{z}{2} = 1 \quad \text{(or x + 4z = 8)}
  2. Formulate the inequalities: Since the shaded region is in the first octant where x0x \geq 0, y0y \geq 0, and z0z \geq 0, we should consider the inequalities that represent the planes and restrict the solution to this region.

    The inequalities representing the region would be:

    • x+2y+4z4x + 2y + 4z \leq 4
    • y+z2y + z \leq 2
    • x+4z8x + 4z \leq 8
    • x0x \geq 0
    • y0y \geq 0
    • z0z \geq 0

Comparing with Options:

Let’s compare this with the provided options:

Option A:

  • x+2y+4z4x + 2y + 4z \leq 4
  • y+z2y + z \leq 2
  • y1-y \leq 1 (This contradicts y0y \geq 0)
  • x0x \geq 0
  • y0y \geq 0
  • z0z \geq 0

Option B:

  • x+2y+4z8x + 2y + 4z \leq 8 (This contradicts our calculation)
  • x+4y+z4x + 4y + z \leq 4 (This is not in our region)
  • y+z2y + z \leq 2
  • x0x \geq 0
  • y0y \geq 0
  • z0z \geq 0

Option C:

  • x+2y+4z4x + 2y + 4z \leq 4
  • y+z2y + z \leq 2
  • x+4z8x + 4z \leq 8
  • x0x \geq 0
  • y0y \geq 0
  • z0z \geq 0

Option D:

  • x+2y+4z8x + 2y + 4z \leq 8 (This contradicts our calculation)
  • y+z2y + z \leq 2
  • x+y+z1x + y + z \leq 1 (This is not in our region)
  • x0x \geq 0
  • y0y \geq 0
  • z0z \geq 0

Conclusion:

The correct option that matches the inequalities representing the shaded region is Option C.

Would you like further details on this solution or have any questions?

Related Questions:

  1. What are the basic steps to derive the equation of a plane given three points?
  2. How do you determine if a point lies inside a given plane in 3D space?
  3. Can you explain how to find the intersection line of two planes?
  4. What is the significance of the first octant in 3D geometry?
  5. How do inequalities in three dimensions differ from those in two dimensions?

Tip:

When analyzing 3D geometric regions, always start by identifying the intercepts and equations of boundary planes. Then, translate these into inequalities that represent the feasible region.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Plane Geometry
3D Coordinate Geometry

Formulas

Equation of a plane in 3D: ax + by + cz = d
Conversion to inequality: ax + by + cz ≤ d

Theorems

Properties of Planes in 3D Geometry
Intersection of Planes

Suitable Grade Level

Grades 11-12