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The given problem asks for the distance between two parallel planes. The equations of the planes are:

1. \( x + 6y + z = 73 \)
2. \( x + 6y + z = 44 \)

To find the distance between two parallel planes, we use the formula:

\[
d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2 + c^2}}
\]

Here, the general form of the plane equation is \( ax + by + cz = d \), and for both planes, \( a = 1 \), \( b = 6 \), and \( c = 1 \).

### Applying the formula:

1. \( c_1 = 73 \), \( c_2 = 44 \)
2. \( a^2 + b^2 + c^2 = 1^2 + 6^2 + 1^2 = 1 + 36 + 1 = 38 \)

Now, calculate the distance:

\[
d = \frac{|73 - 44|}{\sqrt{38}} = \frac{29}{\sqrt{38}}
\]

Approximating:

\[
d \approx \frac{29}{6.1644} \approx 4.7044
\]

Thus, the distance between the two planes is approximately **4.7044**.

Would you like more details on any of these steps?

### Related Questions:
1. How do you find the distance between a point and a plane?
2. What is the geometric interpretation of parallel planes?
3. Can non-parallel planes have a fixed distance between them?
4. What happens if two planes are coincident? What is their distance?
5. How do we compute the angle between two non-parallel planes?

**Tip**: When working with distances between geometric entities, always ensure the direction vectors or coefficients are normalized if necessary to avoid errors.

Find the distance between the following parallel planes: x + 6y + z = 73 and x + 6y + z = 44.

Learn how to calculate the distance between two parallel planes x + 6y + z = 73 and x + 6y + z = 44 using the formula for distance between planes. A detailed step-by-step solution is provided, suitable for students in Grades 10-12.

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