1.

a man travels by boat 36km down a river and back in 8hrs. If the speed of his boat in still water is 12km/hr, find the speed of the river

Answer»

A MAN travels by boat 36 km down a river and back in 8 HRS and the SPEED of his boat in still water is 12km/h

so, Distance, d = 36km

Total TIME taken for down a river and back in 8hrs
so, Time of downstream + Time of upstream = 8.................(1)

Speed of his boat in still water is 12km/hr
Speed of stream or river = x km/h

so, Speed of downstream = speed of boat in still water + speed of river = (12 + x) km/h

Speed of upstream = Speed of boat in still water - speed of river = (12 - x) km/h

so, Time of downstream = Distance / Speed of downstream = 36/(12+x)................(2)

Time of upstream = Distance / speed of upstream = 36/(12-x).......................(3)

substitute (2) and (3) in (1)
\frac{36}{(12 + x)} + \frac{36}{(12 - x)} = 8 \\ \frac{36(12 - x) + 36(12 + x)}{(12 + x)(12 - x)} = 8 \\ \frac{36(12 - x + 12 + x)}{ {12}^{2} - {x}^{2} } = 8 \\ \frac{36(24)}{144 - {x}^{2} } = 8 \\ 864 = 8(144 - {x}^{2} ) \\ 864 = 1152 - 8 {x}^{2} \\ 8 {x}^{2} = 1152 - 864 \\ 8 {x}^{2} = 288 \\ {x}^{2} = \frac{288}{8} \\ {x}^{2} = 36 \\ x = 6
so the speed of the river = 6km/h

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