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To find the distance between two parallel planes, the general formula is:

\[
d = \frac{|c_1 - c_2|}{\sqrt{A^2 + B^2 + C^2}}
\]

where the planes are given in the form \(Ax + By + Cz = D\). In your case, the two planes are:

\[
x + 6y + z = 73 \quad \text{(first plane)}
\]
\[
x + 6y + z = 44 \quad \text{(second plane)}
\]

From these equations, we can see:
- \(A = 1\), \(B = 6\), and \(C = 1\) (since the coefficients of \(x\), \(y\), and \(z\) are the same for both planes).
- \(D_1 = 73\) and \(D_2 = 44\).

Now, applying the formula:

\[
d = \frac{|73 - 44|}{\sqrt{1^2 + 6^2 + 1^2}} = \frac{29}{\sqrt{1 + 36 + 1}} = \frac{29}{\sqrt{38}}.
\]

Let's calculate the result.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the final steps manually.

The expression we need to compute is:

\[
d = \frac{29}{\sqrt{38}}.
\]

You can now evaluate this using a calculator or a math tool by finding \(\sqrt{38}\), and then dividing 29 by that result.

Do you want me to help further with that, or would you like to explore more questions? Here are some related queries to expand your understanding:

1. How can we find the normal vector of a plane in three-dimensional geometry?
2. What does it mean for two planes to be parallel in terms of their normal vectors?
3. How is the distance between non-parallel planes calculated?
4. Can the distance between skew lines be calculated similarly to parallel planes?
5. How does changing the coefficients of a plane's equation affect its orientation in space?

**Tip:** To find the distance between two objects in 3D space, it's often useful to express everything in vector form!

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

Learn how to find the distance between two parallel planes, x + 6y + z = 73 and x + 6y + z = 44, using the distance formula. This solution breaks down the key concepts in analytic geometry and demonstrates the step-by-step calculation. Suitable for Grades 10-12 students.

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