SOLUTION: In still water, Mo and Sally can paddle a canoe at 12 mph. It takes them a total of 6 hours to paddle 32 miles up a river and then return the same distance to their starting point.

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Question 940709: In still water, Mo and Sally can paddle a canoe at 12 mph. It takes them a total of 6 hours to paddle 32 miles up a river and then return the same distance to their starting point. What is the rate of the current?
Found 2 solutions by josmiceli, macston:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+12+ mi/hr is their speed in still water
Let +c+ = the speed of the current in mi/hr
+12+%2B+c+ = their speed in mi/hr going downriver
+12+-+c+ = their speed in mi/hr going upriver
Let +t+ = their time in hours to go upriver
+6+-+t+ = their time in hours to go downriver
------------------------------------------
Equation for going upriver:
(1) +32+=+%28+12+-+c+%29%2At+
Equation for going downriver:
(2) +32+=+%28+12+%2B+c+%29%2A%28+6+-+t+%29+
----------------------------
(1) +32+=+12t+-+c%2At+
(1) +c%2At+=+12t+-+32+
(1) +c+=+12+-+32%2Ft+
and
(2) +32+=+72+%2B+6c+-+12t+-+c%2At+
(2) +40+=+12t+-+6c+%2B+c%2At+
---------------------------
Substitute (1) into (2)
(2) +40+=+12t+-+6%2A%28+12+-+32%2Ft+%29+%2B+%28+12+-+32%2Ft+%29%2At+
(2) +40+=+12t+-+72+%2B+192%2Ft+%2B+12t+-+32+
(2) +144+=+192%2Ft+%2B+24t+
(2) +18+=+24%2Ft+%2B+3t+
(2) +18t+=+24+%2B+3t%5E2+
(2) +3t%5E2+-+18t+%2B+24+=+0+
Use the quadratic formula
+t+=+%28+-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+3+
+b+=+-18+
+c+=+24+
+t+=+%28+-%28-18%29+%2B-+sqrt%28+%28-18%29%5E2+-+4%2A3%2A24+%29%29+%2F+%282%2A3%29+
+t+=+%28+18+%2B-+sqrt%28+324+-+288+%29%29+%2F+6+
+t+=+%28+18+%2B-+sqrt%28+36+%29%29+%2F+6+
+t+=+%28+18+%2B+6+%29+%2F+6+
+t+=+24%2F6+
+t+=+4+ hrs ( I picked the right square root because
the trip upriver should take longer than downriver )
+6+-+t+=+6+-+4+
+6+-+t+=+2+ hrs
-------------------
(1) +32+=+%28+12+-+c+%29%2At+
(1) +32+=+%28+12+-+c+%29%2A4+
(1) +8+=+12+-+c+
(1) +c+=+4+
The rate of the current is 4 mi/hr
------------------------------
check:
(2) +32+=+%28+12+%2B+c+%29%2A%28+6+-+t+%29+
(2) +32+=+%28+12+%2B+4+%29%2A%28+6+-+4+%29+
(2) +32+=+16%2A2+
(2) +32+=+32+
OK



Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
First they went 32miles against the current given by:
32miles%2F%2812+mph-C%29=Time+upstream where c is the speed of the current and 12mph is their normal speed
Then downstream given by:
32miles%2F%2812+mph%2BC%29=Time+downstream
And for the total trip
32miles%2F%2812+mph-C%29%2B+32miles%2F%2812+mph%2BC%29=6+hours Multiply both sides by 12mph-C
12mph-C%2832miles%2F%2812+mph-C%29%2B+32miles%2F%2812+mph%2BC%29%29=%286+hours%2912mph-C

32miles+%2B+%2832+miles%29%2812mph+-+C%29+%2F%2812+mph+%2BC%29=6+Hrs%2812mph-C%29Multiply both sides by (12mph+C)

32+miles%2812+mph%2BC%29%2B+%2832+miles%29%2812+mph-C%29=6+hrs%2812mph%2BC%29%2812mph-C%29
+%2832+miles%29%28%2812mph%2BC%29%2B%2812mph-C%29%29=6+HRS%2812mph%2BC%29%2812mph-C%29
+%2832+miles%29%2824+mph%29=6+Hrs%28144-c%5E2%29Divide both sides by 6
%28%2832+miles%29%2824mph%29%29%2F6=%286+hrs%28144-c%5E2%29%29%2F6
+%2832+miles%29%284+mph%29=144-c%5E2
128=144-c%5E2 subtract 128 from both sides
+0=16-c%5E2Add c%5E2+ to both sides
c%5E2=16 so c = plus/minus 4 So the current is 4 MPH
CHECK:
upstream + downstream= 6 hours
upstream=32+miles%2F%2812mph-4mph%29=32+miles%2F8+mph+=+4+hrs
downstream=%2832+miles%29%2F%2812mph%2B4mph%29=32+miles%2F16+mph=+2+hrs

2Hrs + 4hrs= 6hrs