A drone must deliver a medical kit from location D to a remote clinic at location E. The route consists of two regions:
• Region I: Over level ground where the maximum drone speed (relative to the ground) is a constant 15 m/s.
• Region II: Through a mountain pass where strong headwinds reduce its maximum ground speed to 9 m/s.
The straight-line distance from D to the entrance M of the mountain pass is 2400 m, and the straight-line distance from M to E is 2000 m. Instead of following the straight path DM, the drone may divert at an angle to point P on the boundary between the two regions before entering the pass. Assume the boundary between regions is a straight line perpendicular to DM at M.
Let the perpendicular distance from M to P be h meters (so h=0 at M, and P lies north of the line DM). The segment DP then has length 24002+h2, and the segment PE through the pass has length 20002+h2. Ignore acceleration and assume instantaneous change of speed at P.
(a) Derive an expression for the total delivery time T(h) in terms of h.
(b) Find the value of h (to the nearest meter) that minimizes T(h). Provide calculus-based justification that this value yields a minimum.
(c) Compute the corresponding minimum delivery time, in seconds, rounded to the nearest whole number.
(d) Discuss how the optimal path would change if the headwind in Region II intensified so that the maximum speed there dropped below 9 m/s. No calculation is required; base your explanation on the structure of T(h).