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The problem asks to find the distance from the point \( (8, 0, 0) \) to the line given by the parametric equations:

\[
x = 8 + t, \quad y = 2t, \quad z = 6\sqrt{5} + 2t.
\]

### Step 1: Identify the direction vector of the line.
From the parametric equations, the direction vector \( \mathbf{v} \) of the line is the coefficients of \( t \) in each equation:

\[
\mathbf{v} = (1, 2, 2).
\]

### Step 2: Define the point on the line.
At \( t = 0 \), the point on the line is:

\[
P_0 = (8, 0, 6\sqrt{5}).
\]

### Step 3: Use the formula for the distance from a point to a line.
The formula for the distance from a point \( P_1 \) to a line passing through a point \( P_0 \) with direction vector \( \mathbf{v} \) is:

\[
d = \frac{\left| \mathbf{AP_0} \times \mathbf{v} \right|}{|\mathbf{v}|},
\]

where \( \mathbf{AP_0} \) is the vector from the point \( P_1 \) to the point \( P_0 \).

- \( P_1 = (8, 0, 0) \) (the point we are calculating the distance from),
- \( P_0 = (8, 0, 6\sqrt{5}) \),
- \( \mathbf{v} = (1, 2, 2) \).

### Step 4: Compute the vectors and the cross product.
Now, I'll calculate the distance using these parameters.The distance from the point \( (8, 0, 0) \) to the given line is approximately \( 10 \) units. This matches the correct answer from the choices provided.

Would you like further details on any step?

Here are 5 related questions for further practice:
1. How do you find the shortest distance between two skew lines?
2. What is the distance from a point to a plane?
3. How do you derive the equation of a plane given three points?
4. How is the dot product used in finding angles between vectors?
5. What is the geometric interpretation of the cross product?

**Tip:** In 3D geometry, the direction vector of a line plays a crucial role in determining distances and angles between lines and planes.

Find the distance from the point (8, 0, 0) to the line: x = 8 + t, y = 2t, z = 6√5 + 2t.

This problem involves finding the distance between the point (8, 0, 0) and a line with parametric equations in 3D space. It uses vector operations like the cross product and line direction vector to calculate the shortest distance.

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