Final answer:
To find the value of k that makes AC + BC as small as possible, we need to calculate the distance between points A and C and the distance between points B and C, and then find the value of k that minimizes the sum of these distances.
Step-by-step explanation:
The question asks for the value of k that makes AC + BC as small as possible. The coordinates of points A, B, and C are given as (5, 5, k), (2, 1, k), and (0, k). To find the value of k, we need to calculate the distance between points A and C and the distance between points B and C, and then find the value of k that minimizes the sum of these distances.
The distance between two points in 3D space can be calculated using the formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
Using this formula, we can calculate the distance between points A and C as: dAC = sqrt((5 - 0)^2 + (5 - k)^2 + (k - 0)^2).
Similarly, the distance between points B and C is: dBC = sqrt((2 - 0)^2 + (1 - k)^2 + (k - 0)^2).
To find the value of k that minimizes the sum of these distances, we need to minimize the function: f(k) = dAC + dBC.
We can find the minimum value of this function by taking the derivative of f(k) with respect to k and setting it equal to zero. Once we find the value of k that satisfies this equation, we can substitute it back into f(k) to find the minimum value.
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