```Question 100322
A man in a rowboat rows 2 kilometres upstream when at this point a sudden wind blows his hat into the river. He ignores this slight inconvenience and continues to row upstream for another 15 minutes(1/4 hr)
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At this point he turns around and paddles downstream at the same rate he had paddled upstream. Upon reaching the point where he entered the river he noticed he had caught up with his hat. From all of this dilemma the question is, "At what rate is the river flowing'?
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I assume because of the phrase "another 15 min" that he rowed the 2 kilometers
from the start in 15 min, then rowed another 15 min and 2 more kilometers
At that point he will have rowed for 1/2 hr and was 4 km from the start.
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Let s = the speed of the boat in still water
Let x = speed of the current
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Since he travels 4 km in half an hour up-stream, effective speed = 8 km/h
we can say:
s - x = 8
or
s = (x + 8)
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The boat travels up-stream 2 km at 8 km (s - x)
s = (x+8)
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speed down stream:
s + x
Substitute (x+8) for s and you have
(x + 8) + x = (2x + 8)
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Then it goes down-stream 4 km at (8 + 2x)
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Meanwhile the hat travels down stream 2 km at x speed
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Write a time equation: Time = dist/speed
boat time up + boat time down = hat floating time
{{{2/8}}} + {{{4/(8+2x)}}} = {{{2/x}}}
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Solve this equation for x, multiply eq by 8x(8+2):
2(x(8+2x)) + 4(8x) = 2(8(8+2x))
16x + 4x^2 + 32x = 128 + 32x
4x^2 + 16x + 32x - 32x - 128 = 0
4x^2 + 16x - 128 = 0
Simplify, divide by 4:
x^2 + 4x - 32 = 0
Factors to:
(x + 8)(x - 4) = 0
x = +4, it's the positive solution we want here.
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Current is 4 km/h
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You can prove this in the original problem
Boat speed was s - x = 8
s - 4 = 8
s = 12 is the boat speed in still water.
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it went 2 km at 8 and 4 km at 16 while the hat went 2 km at 4
{{{2/8}}} + {{{4/16}}} = {{{2/4}}}
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Did this make sense? Hope so.

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