SOLUTION: a small motorboat travels 14mph in still water. It takes 6 hours longer to travel 59 miles going upstream than it does down stream. Find the rate of the current.

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Question 545887: a small motorboat travels 14mph in still water. It takes 6 hours longer to travel 59 miles going upstream than it does down stream. Find the rate of the current.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let w = rate of current
Let b = rate of boat in still water
+b+-+w+ = rate going against current
+b+%2B+w+ = rate going with current
Let t = time for boat to go 59 mi with current
-------------
given:
equation for going downstream ( with current )
+59+=+%28+b+%2B+w+%29%2A+t+
+59+=+%28+14+%2B+w+%29%2At+
(1) +t+=+59+%2F+%28+14+%2B+w+%29+
equation for going upstream ( against current )
+59+=+%28+b+-+w+%29%2A%28+t+%2B+6+%29+
+59+=+%28+14+-+w+%29%2A%28t+%2B+6+%29+
+59+=+14t+-+w%2At+%2B+84+-+6w+
(2) +t%2A%28+w+-+14+%29+%2B+6w+=+84+-+59+
(2) +t%2A%28+w+-+14+%29+%2B+6w+=+25+
------------------
Substitute (1) into (2)
(2) +%28+59%2F%28+14+%2B+w+%29%29%2A%28+w+-+14+%29+%2B+6w+=+25+
multiply both sides by +14+%2B+w+%29+
(2) +59%2A%28+w+-+14+%29+%2B+6c%2A%28+14+%2B+w+%29+=+25%2A%28+14+%2B+w+%29+
(2) +59w+-+826+%2B+84w+%2B+6w%5E2+=+350+%2B+25w+
(2) +6w%5E2+%2B+84w+%2B+59w+-+25w+-+825+-+350+=+0+
(2) +6w%5E2+%2B+118w+-+1175+=+0+
Use quadratic equation
w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+6+
+b+=+118+
+c+=+-1175+
w+=+%28-118++%2B-+sqrt%28+118%5E2-4%2A6%2A%28-1175%29+%29%29%2F%282%2A6%29+
w+=+%28-118++%2B-+sqrt%28+13924+%2B+28200+%29%29+%2F+12+
w+=+%28-118++%2B-+sqrt%28+42124+%29%29+%2F+12+
+w+=+%28+-118+%2B+205.24%29+%2F+12+
+w+=+87.24%2F12+
+w+=+7.27+
The rate of the current is 7.27 mi/hr
check answer:
+59+=+%28+14+%2B+w+%29%2At+
+59+=+%28+14+%2B+7.27+%29%2At+
+59+=+21.27t+
+t+=+2.774+
and
+59+=+%28+14+-+w+%29%2A%28t+%2B+6+%29+
+59+=+%28+14+-+7.27+%29%2A%28t+%2B+6+%29+
+59+=+6.73%2A%28+t+%2B+6+%29+
+59+=+6.73t+%2B+40.38+
+6.73t+=+59+-+40.38+
+6.73t+=+18.62+
+t+=+2.767+
This looks close enough