Find the equation of the surface.
A standard cone with vertex at (5, 0 , 0) opening up on the positive x-axis.

To find the equation of a surface for a cone with a vertex at \((5, 0, 0)\) and opening along the positive \(x\)-axis, we start by analyzing the geometric properties of the cone.

### Step-by-Step Solution

1. **Standard Cone Equation:**
   The standard equation of a cone with its vertex at the origin and opening along the \(x\)-axis is given by:
   \[
   x^2 = y^2 + z^2.
   \]

2. **Transform the Vertex:**
   Since the vertex is at \((5, 0, 0)\), we shift the coordinate system so that the new vertex is at this point. To do this, we substitute \(x' = x - 5\), \(y' = y\), and \(z' = z\).

   Substituting these transformations into the standard cone equation, we get:
   \[
   (x' + 5)^2 = y'^2 + z'^2.
   \]

3. **Simplify and Replace the Variables:**
   Replace \(x'\), \(y'\), and \(z'\) with \(x\), \(y\), and \(z\), respectively:
   \[
   (x - 5)^2 = y^2 + z^2.
   \]

4. **Final Equation:**
   The equation of the cone with its vertex at \((5, 0, 0)\) and opening along the positive \(x\)-axis is:
   \[
   (x - 5)^2 = y^2 + z^2.
   \]

### Interpretation

This equation represents a cone with its vertex at the point \((5, 0, 0)\), and it opens symmetrically along the positive \(x\)-axis. The cross-sections perpendicular to the \(x\)-axis are circles whose radius increases as \(x\) moves away from 5 in the positive direction.

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### Related Questions:

1. How would the equation change if the cone opened along the negative \(x\)-axis instead?
2. What is the general form of a cone equation in 3D space?
3. How do we find the surface area of a cone given this equation?
4. Can this equation be converted to a cylindrical coordinate system?
5. How does this cone's equation change if the cone is rotated about another axis?

#### Tip: 
Transforming the coordinates is a useful technique when shifting or rotating geometric shapes in 3D space!

Find the equation of the surface. A standard cone with vertex at (5, 0 , 0) opening up on the positive x-axis.

Explore the step-by-step process to find the equation of a cone with its vertex at (5, 0, 0) and opening along the positive x-axis. This problem covers key concepts in 3D geometry and conic sections, making it ideal for students in Grades 10-12.

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