Question 9: Distance Between Points A and C A boat takes a total of 16 hours for traveling...
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Question 9: Distance Between Points A and C
A boat takes a total of 16 hours for traveling downstream from point A to point B and coming back to point C, which is somewhere between A and B. If the speed of the boat in still water is 9 km/hr and the rate of stream is 6 km/hr, what is the distance between A and C?
Show answer & explanation
The problem provides the total time (16 hours) and the speeds, but does not specify the location of point C or the distance from A to B. Without knowing either the distance AC or the distance AB, we cannot determine a unique value for AC. The system is underdetermined: for any valid distance AC ≤ AB, we can find a corresponding distance AB that satisfies the 16-hour constraint.
Step-by-step Derivation:
Let b = speed of boat in still water = 9 km/hr, s = speed of stream = 6 km/hr. Downstream speed = 9 + 6 = 15 km/hr; Upstream speed = 9 - 6 = 3 km/hr. Let AB = x km and AC = y km (where 0 < y ≤ x). Time equation: x/15 + y/3 = 16. Rearranging: x/15 + 5y/15 = 16 → x + 5y = 240 → x = 240 - 5y. For any y in the valid range, x is determined, but y itself is free. For example: if y = 30, then x = 90 (checking: 90/15 + 30/3 = 6 + 10 = 16 ✓); if y = 60, then x = 60 (checking: 60/15 + 60/3 = 4 + 20 = 24 ✗ invalid); if y = 42, then x = 30 (checking: 30/15 + 42/3 = 2 + 14 = 16 ✓). Multiple solutions exist, so AC cannot be uniquely determined.