What is the rate of the river's current if a tug tows a barge 24 miles up a river at 10 miles per hour and returns down the river at 12 miles per hour The trip took 5 and a half hours?

Answer 1

Assuming the tug speed is relative to the river flow and is constant over each trip, and the river flows at the same constant rate during the trips, then:

Let the flow rate down river be x mph

Then the land speed when going upriver is 10 - x mph

and the land speed when going down river is 12 + x mph

This gives the total time for the journeys:

time = distance/speed

→ 5½ hours = 24 miles/(10 - x)mph + 24 miles/(12 + x)mph

→ 11/2 = 24(12 + x + 10 - x)/(10 - x)(12 + x)

→ 11/2 = 24×22/(120 - 2x - x²)

→ 120 - 2x - x² = 24×22×2/11

→ x² + 2x - 120 + 96 = 0

→ x² + 2x - 24 = 0

→ (x - 4)(x + 6) = 0

→ x = 4 or -6

→ The river is flowing at 4 mph down the river or 6 mph up the river.

If the river is non-tidal, then the river is flowing at 4 mph down the river.

(In river terms, up is towards the source of the river, down is towards the mouth of the river, and for a tidal river, eg the Thames between Richmond and its mouth, the current can be flowing up or down the river - the rate will not be constant, but for a period of time whilst the tide is flowing in one direction can be considered to be constant.)