You are a member of a geological team in Central Africa. Your team comes upon a

Question

You are a member of a geological team in Central Africa. Your team comes upon a wide river that is flowing east. You must determine the width of the river and the current speed (the speed of the water relative to the earth). You have a small boat with an outboard motor. By measuring the time it takes to cross a pond where the water isnt flowing, you have calibrated the throttle settings to the speed of the boat in still water. You set the throttle so that the speed of the boat relative to the river is a constant 6.00 m/s.Traveling due north across the river, you reach the opposite bank in 20.1 s. For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 7.00 m/s . You travel due south from one bank to the other and cross the river in 11.2 s.

How wide is the river?

Explanation / Answer

suppose current speed is v.
when going north,

v = 6sin@1 (@ is angle made by boat with current)

along the river width ,

vy = 6cos@1

and 6 cos@1 x 20.1 = d

when going south,

v = 8.20sin@2

similarly 8.20cos@2 x 11.2 = d

so 6 sin@1 = 8.20sin@2 and 120.6cos@1 = 91.84cos@2

36 sin^2@1 = 67.24 sin^2 @2

36 (1 - cos^2 @1 ) = 67.24 sin^2 @2

1 - (91.84 cos@2 / 120.6)^2 = 1.868 ( 1 - cos^2 @2)

1.288 cos^2 @2 = 0.868

@2 = 34.82 deg

6 sin@1 = 8.20sin34.82

@1 = 51.29 deg

1) width = 8.20cos@2 x 11.2 = 91.84 x cos34.82 = 75.40 m

2) v = 8.20 sin@2 = 4.68 m/s

3) for the shortest time , speed of boat must be perpendicular to the current.

t = 75.40/6 = 12.57 s

4) distance = vt = 4.68 x 12.57 = 58.81 m

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