Final answer:
The length of the shortest path from (0, 0) to (12, 16) that does not go inside the circle (x - 6)² + (y - 8)² = 25 is 8.94. The correct options is (D).
Step-by-step explanation:
To find the length of the shortest path from (0, 0) to (12, 16) that does not go inside the circle (x - 6)² + (y - 8)² = 25, we can first find the shortest distance between the two points and then subtract the length of the segment inside the circle.
The shortest distance between the two points is the length of the straight line connecting them, which can be found using the distance formula.
D = √((x2 - x1)² + (y2 - y1)²) = √((12 - 0)² + (16 - 0)²) = √(144 + 256) = √400 = 20
The length of the segment inside the circle can be found by finding the length of the arc of the circle subtended by the straight line connecting the two points.
θ = 2 * tan⁻¹((d/2) / r) = 2 * tan⁻¹((10 / 5) = 2 * tan⁻¹(2) ≈ 2 * 63.4° = 126.8°
Length of segment inside circle = (θ/360°) * 2πr = (126.8/360) * 2 * π * 5 = (0.352 * 10 * π) ≈ 11.06
Therefore, the length of the shortest path is 20 - 11.06 = 8.94. So, option (D) is the correct answer.