A boat takes 128 minutes less to travel 48 km downstream than to travel the same distance...
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A boat takes 128 minutes less to travel 48 km downstream than to travel the same distance upstream. If the speed of the stream is 3 km/hr, then the speed of the boat in still water is?
Show answer & explanation
The problem asks for the speed of the boat in still water given the time difference between upstream and downstream travel for a fixed distance. By setting up the time difference equation using the formula Time = Distance/Speed, the speed of the boat is found to be 12 km/hr.
Step-by-step Derivation:
Step 1: Identify given values. Distance (D) = 48 km, Speed of stream (v) = 3 km/hr, Time difference (Δt) = 128 minutes = 128/60 hours = 32/15 hours.
Step 2: Let the speed of the boat in still water be 's'.
Step 3: Downstream speed = s + 3; Upstream speed = s - 3.
Step 4: Set up the equation for time difference: Time(upstream) - Time(downstream) = Δt
48/(s - 3) - 48/(s + 3) = 32/15
Step 5: Simplify the equation: 48 * [(s + 3 - (s - 3)) / ((s - 3)(s + 3))] = 32/15
48 * [6 / (s^2 - 9)] = 32/15
Step 6: Solve for s: 288 / (s^2 - 9) = 32/15
(288 * 15) / 32 = s^2 - 9
9 * 15 = s^2 - 9
135 = s^2 - 9
144 = s^2
Step 7: s = √144 = 12 km/hr.