we need to calculate the distance between the surface:
9⋅z^2−54⋅z+9⋅y^2−54⋅y+81⋅x^2−162⋅x=−162
and the plain x=6

To find the distance between the given surface and the plane \(x = 6\), we need to follow these steps:

### Step 1: Simplify the Equation of the Surface
The equation of the surface is:

\[
9z^2 - 54z + 9y^2 - 54y + 81x^2 - 162x = -162
\]

We can simplify this by completing the square for each variable term.

**For \(z\):**
\[
9(z^2 - 6z) = 9\left[(z - 3)^2 - 9\right] = 9(z - 3)^2 - 81
\]

**For \(y\):**
\[
9(y^2 - 6y) = 9\left[(y - 3)^2 - 9\right] = 9(y - 3)^2 - 81
\]

**For \(x\):**
\[
81(x^2 - 2x) = 81\left[(x - 1)^2 - 1\right] = 81(x - 1)^2 - 81
\]

Substituting these back into the surface equation:

\[
9(z - 3)^2 - 81 + 9(y - 3)^2 - 81 + 81(x - 1)^2 - 81 = -162
\]

Simplify:

\[
9(z - 3)^2 + 9(y - 3)^2 + 81(x - 1)^2 = -162 + 243 = 81
\]

Divide by 81:

\[
\frac{(z - 3)^2}{9} + \frac{(y - 3)^2}{9} + (x - 1)^2 = 1
\]

This is the equation of an ellipsoid centered at \((1, 3, 3)\) with semi-axes lengths \(1\), \(3\), and \(3\).

### Step 2: Calculate the Distance from the Center of the Ellipsoid to the Plane \(x = 6\)

The plane \(x = 6\) is parallel to the \(yz\)-plane. The center of the ellipsoid is at \((1, 3, 3)\).

The distance \(d\) from the point \((1, 3, 3)\) to the plane \(x = 6\) is given by:

\[
d = |6 - 1| = 5
\]

### Step 3: Consider the Radius Along the \(x\)-Axis
The ellipsoid has a semi-axis length of \(1\) along the \(x\)-axis. So, the closest point on the ellipsoid to the plane \(x = 6\) along the \(x\)-axis will be \(1 + 1 = 2\) units away from the center.

### Step 4: Subtract the Semi-Axis Length from the Distance
The actual distance between the ellipsoid and the plane \(x = 6\) is:

\[
\text{Distance} = 5 - 1 = 4
\]

So, the distance between the surface and the plane \(x = 6\) is **4 units**.

Would you like to delve deeper into any particular step or have further questions?

---

Here are 8 follow-up questions that might interest you:
1. How do you complete the square for quadratic expressions in multiple variables?
2. What are the general properties of an ellipsoid, and how do they differ from those of a sphere?
3. How do you find the distance from a point to a plane in three-dimensional space?
4. Can you explain the geometric meaning of the semi-axes in an ellipsoid?
5. What happens if the given plane were not parallel to one of the coordinate axes?
6. How do you determine the shortest distance from an arbitrary point to an ellipsoid?
7. How would the process change if the surface were a hyperboloid instead of an ellipsoid?
8. How do transformations like translations and rotations affect the equation of surfaces like ellipsoids?

**Tip:** When dealing with distances in geometry, it's often helpful to visualize the shapes and their relative positions in 3D space to better understand the problem.

Learn how to calculate the distance between a given ellipsoid surface and the plane x = 6 in three-dimensional space. This problem involves geometric concepts such as ellipsoids, distances in 3D space, and specific calculations using distance formula. Suitable for advanced mathematics students and professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation