The problem asks to find the distance traveled by the bus, where the bus starts from point $A(-4, 3, 2)$ and moves to point $B(1, -2, 7)$.

To find the distance between two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in three-dimensional space, we use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Substituting the given points $A(-4, 3, 2)$ and $B(1, -2, 7)$:

$d = \sqrt{(1 - (-4))^2 + (-2 - 3)^2 + (7 - 2)^2}$

Simplifying the expression:

$d = \sqrt{(1 + 4)^2 + (-2 - 3)^2 + (7 - 2)^2}$ $d = \sqrt{5^2 + (-5)^2 + 5^2}$ $d = \sqrt{25 + 25 + 25}$ $d = \sqrt{75}$ $d = 5\sqrt{3}$

Thus, the distance traveled by the bus is $5\sqrt{3}$ units.

Would you like more details on this solution or have any questions?

Here are five related questions:

1. What is the formula for finding the distance between two points in 2D?
2. How do you calculate the midpoint between two points in 3D?
3. What is the equation of a line passing through two points in 3D space?
4. How do you generalize the distance formula for $n$-dimensional space?
5. How does the Pythagorean theorem apply to finding distance in 3D?

Tip: Always double-check signs when subtracting coordinates to avoid calculation mistakes.