SOLUTION: There are 2 boats docked on opposite sides of a river. The sides of the river are parallel and the current is negligible. The 2 boats set out to cross the river at the same time, a

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Question 163569: There are 2 boats docked on opposite sides of a river. The sides of the river are parallel and the current is negligible. The 2 boats set out to cross the river at the same time, and each maintains a constant rate. They first meet at 800 yards from side 1, then proceed to the opposite side and turn around. Next they meet 300 yards from side 2. How wide is the river? What happens if 800 yards is replaced with D1 and 300 yards is replaced with D2 if both D1 and D2 are assumed to be less than the width of the river?
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There are 2 boats docked on opposite sides of a river. The sides of the river are parallel and the current is negligible. The 2 boats set out to cross the river at the same time, and each maintains a constant rate. They first meet at 800 yards from side 1, then proceed to the opposite side and turn around. Next they meet 300 yards from side 2. How wide is the river?
:
Let d = width of the river
:
When they meet the 1st time
Boat 1 travels 800 yds
Boat 2 travels (d-800) yds
:
When they meet the 2nd time:
Boat 1 has traveled (d-800) + 300 = d-500 yds
Boat 2 has traveled 800 + (d-300) = d+500 yds
:
relationship of boat 1 dist to boat 2 dist should be the same to both meetings:
800%2F%28%28d-800%29%29 = %28%28d-500%29%29%2F%28%28d%2B500%29%29
Cross multiply:
800(d+500) = (d-800)(d-500)
:
800d + 400000 = d^2 - 1300d + 400000
:
0 = d^2 - 1300d - 800d + 400000 - 400000
:
d^2 - 2100d = 0
:
d(d - 2100) = 0
:
d = 2100 yds across
:
:
What happens if 800 yards is replaced with D1 and 300 yards is replaced with D2 if both D1 and D2 are assumed to be less than the width of the river?
:
Let d = width of the river
:
When they meet the 1st time
Boat 1 travels d1 yds
Boat 2 travels (d-d1) yds
:
When they meet the 2nd time:
Boat 1 has traveled (d-d1) + d2 yds
Boat 2 has traveled d1 + (d-d2) yds
:
relationship of boat 1 dist to boat 2 dist should be the same to both meetings:
d1%2F%28%28d-d1%29%29%29 = %28%28d-d1%2Bd2%29%29%2F%28%28d%2Bd1-d2%29%29