Bond (007) wants to move from a point A to point D, but here 2 watchtowers (B…
2025
Bond (007) wants to move from a point A to point D, but here 2 watchtowers (B & C) are present in the way of A to D. soldier on watchtowers has rifles with range of 1 kilometer. A,B,C,D are in straight line and distance between All points is 1 kilometer respectively. Find the minimum distance covered by 007 in journey between point A to D while his distance from watchtowers is always >=1.
- A.
3
- B.
4
- C.
3.14
- D.
4.14
Attempted by 244 students.
Show answer & explanation
Correct answer: D
Answer: minimum distance = 1 + π ≈ 4.14 km
Reasoning:
Set coordinates along the straight line: let A = (0,0), B = (1,0), C = (2,0), D = (3,0). Each watchtower has radius 1 km (forbidden interior). A and D lie on the boundaries of the first and second circles respectively.
By symmetry, the shortest path that stays at distance ≥1 from both watchtowers goes above (or below) the line and is composed of:
A quarter-circle arc along the boundary of the first watchtower from A to its top tangent point. Arc length = (π/2)·1 = π/2.
A straight horizontal segment between the two top tangent points (from (1,1) to (2,1)), length = 1 km.
A quarter-circle arc along the boundary of the second watchtower from its top tangent point down to D. Arc length = π/2.
Total length = π/2 + 1 + π/2 = 1 + π ≈ 4.1416 km.
Why this is minimal: any valid path must remain outside the circular forbidden regions. Between the two boundary arcs the straight segment is the shortest connector, and following the circle boundary between fixed boundary points is the shortest route constrained to the circle. Symmetry ensures the top (or bottom) symmetric path is optimal.