Final answer:
The shortest distance from the point (5, 0, −6) to the plane x + y + z = 6 is calculated using a standard formula and is approximately 4.04 units.
Step-by-step explanation:
The question asks to find the shortest distance, d, from a point to a plane in three-dimensional space. This is a common problem in geometry and algebra. To calculate the shortest distance from a point to a plane, we use the formula:
d = |Ax + By + Cz + D| / √(A² + B² + C²)
where (x, y, z) is the point in question, and Ax + By + Cz + D = 0 is the equation of the plane. In this case, the point is (5, 0, −6) and the plane's equation is x + y + z = 6. Substituting these values into the formula, we get:
d = |(1)(5) + (1)(0) + (1)(−6) + (−6)| / √(1² + 1² + 1²) = 7 / √3
d ≈ 4.04 units