The distance between the parallel planes 10x + 2y + 2z = 10 and 5x + y + Z = 2 is

To find the distance between the two parallel planes given by the equations:

1. \(10x + 2y + 2z = 10\)
2. \(5x + y + z = 2\)

we can first rewrite them in the standard form \(Ax + By + Cz = D\).

The first plane can be simplified:
\[
10x + 2y + 2z = 10 \implies 5x + y + z = 5
\]

Now, we have the equations of the two planes:
1. \(5x + y + z = 5\)
2. \(5x + y + z = 2\)

Since the normal vectors of both planes are the same, they are indeed parallel. The distance \(d\) between two parallel planes given by \(Ax + By + Cz = D_1\) and \(Ax + By + Cz = D_2\) can be calculated using the formula:
\[
d = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}}
\]

In this case:
- \(A = 5\)
- \(B = 1\)
- \(C = 1\)
- \(D_1 = 5\)
- \(D_2 = 2\)

Substituting these values into the formula:
\[
d = \frac{|2 - 5|}{\sqrt{5^2 + 1^2 + 1^2}} = \frac{|-3|}{\sqrt{25 + 1 + 1}} = \frac{3}{\sqrt{27}} = \frac{3}{3\sqrt{3}} = \frac{1}{\sqrt{3}}
\]

Thus, the distance between the two parallel planes is:
\[
d = \frac{1}{\sqrt{3}} \text{ units.}
\]

### Summary
The distance between the planes is \(\frac{1}{\sqrt{3}}\).

Do you want further details or explanations? Here are some related questions you might consider:

1. How can you derive the equation of a plane from a point and a normal vector?
2. What is the geometric interpretation of parallel planes in three-dimensional space?
3. Can you show how to derive the distance formula for non-parallel planes?
4. What are some applications of finding distances between planes in real-world scenarios?
5. How do you find the intersection of two planes in three-dimensional space?

**Tip:** When working with planes, always check if they are parallel by comparing their normal vectors.

Discover how to calculate the distance between the parallel planes defined by the equations 10x + 2y + 2z = 10 and 5x + y + z = 2. This step-by-step solution utilizes the distance formula for parallel planes, making it suitable for high school students in Grades 10-12.

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