SOLUTION: The current of a river is 2 miles per hour. It takes a motorboat a total of 3 hours to travel 8 miles upstream and return 8 miles downstream. What is the speed of the boat in still

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: The current of a river is 2 miles per hour. It takes a motorboat a total of 3 hours to travel 8 miles upstream and return 8 miles downstream. What is the speed of the boat in still      Log On


   

Question 1157547: The current of a river is 2 miles per hour. It takes a motorboat a total of 3 hours to travel 8 miles upstream and return 8 miles downstream. What is the speed of the boat in still water?
Found 3 solutions by Boreal, josmiceli, MathTherapy:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
speed of motorboat in still water=s
s-2 is speed upstream and if t hours, the distance is t(s-2)=8, and t=8/(s-2)
s+2 is speed downstream and it is 3-t hours, and the distance is (s+2)(3-t)=8, so 3-t=8/(s+2), t=3-(8/(s+2)
t=(3s+6-8)/(s+2)
so (3s+6-8)/(s+2)=8/(s-2)
cross-multiply
3s^2-6s-2s+4=8s+16
3s^2-16s-12=0
(3s+2)(s-6)=0
s=6 mph speed of boat in still water.
upstream at 4 mph takes 2 hours
downstream at 8 mph takes 1 hour


Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+c+=+2+ mi/hr is the speed of the current
Let +s+ = the speed of the boat in still water
Let +t+ = time in hrs for the upstream trip
+3+-+t+ = time in hrs for the downstream trip
-------------------------------------------------------
Equation for going upstream:
(1) +8+=+%28+s+-+c+%29%2At+
equation for going downstream:
(2) +8+=+%28+s+%2B+c+%29%2A%28+3+-+t+%29+
-------------------------------------
(1) +8+=+%28+s+-+2+%29%2At+
(2) +8+=+%28+s+%2B+2+%29%2A%28+3+-+t+%29+
-----------------------------------
(1) +t+=+8%2F+%28+s+-+2+%29+
Plug this into (2)
(2) +8+=+%28+s+%2B+2+%29%2A%28+3+-+8%2F%28+s-2+%29%29+
(2) +8+=+3s+%2B+6+-+%28+8s+%29+%2F+%28+s-2+%29+-+16+%2F+%28+s-2+%29+
(2) +2+=+3s+-+%28+8s+-+16+%29+%2F+%28+s-2+%29+
Multiply both sides by +s+-+2+
(2) +2%2A%28+s+-+2+%29+=+3s%2A%28+s-2+%29+-+8s+-+16+
(2) +2s+-+4+=+3s%5E2+-+6s+-+8s+-+16+
(2) +3s%5E2+-+16s+-+12+=+0+
Use quadratic formula
+s+=+%28+-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+3+
+b+=+-16+
+c+=+-12+
+s+=+%28+-%28-16%29+%2B-+sqrt%28+%28-16%29%5E2+-+4%2A3%2A%28-12%29+%29%29%2F%282%2A3%29+
+s+=+%28+16+%2B-+sqrt%28+256+%2B+144+%29%29+%2F+6+
+s+=+%28+16+%2B-+sqrt%28+400+%29%29+%2F+6+
+s+=+%28+16+%2B+20+%29+%2F+6+ ( can't use the negative solution )
+s+=+6+
The speed of the boat in still water is 6 mi/hr
--------------------
check:
(1) +8+=+%28+s+-+c+%29%2At+
(1) +8+=+%28+6+-+2+%29%2At+
(1) +8+=+4t+
(1) +t+=+2+ hrs
and
(2) +8+=+%28+s+%2B+c+%29%2A%28+3+-+t+%29+
(2) +8+=+%28+6+%2B+2+%29%2A%28+3+-+t+%29+
(2) +8+=+8%2A%28+3+-+t+%29+
(2) +8+=+24+-+8t+
(2) +8t+=+16+
(2) +t+=+2+ hrs
OK

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The current of a river is 2 miles per hour. It takes a motorboat a total of 3 hours to travel 8 miles upstream and return 8 miles downstream. What is the speed of the boat in still water?
Let speed of boat in still water be S

Then we get the following TIME equation: matrix%281%2C3%2C+3%2C+%22=%22%2C+8%2F%28S+-+2%29+%2B+8%2F%28S+%2B+2%29%29

3(S - 2)(S + 2) = 8(S + 2) + 8(S - 2) ------- Multiplying by LCD, (S - 2)(S + 2) 

matrix%281%2C3%2C+3%28S%5E2+-+4%29%2C+%22=%22%2C+8S+%2B+16+%2B+8S+-+16%29

matrix%281%2C3%2C+3S%5E2+-+12%2C+%22=%22%2C+16S%29

matrix%281%2C3%2C+3S%5E2+-+16S+-+12%2C+%22=%22%2C+0%29

matrix%281%2C3%2C+3S%5E2+-+18S+%2B+2S+-+12%2C+%22=%22%2C+0%29

3S(S - 6) + 2(S - 6) = 0____(S - 6)(3S + 2) = 0

Speed in still water, or highlight_green%28matrix%281%2C4%2C+S%2C+%22=%22%2C+6%2C+mph%29%29               OR               matrix%281%2C4%2C+S%2C+%22=%22%2C+-+2%2F3%2C+mph%29(ignore) 

As you probably noticed, you DON'T need to use the quadratic equation as the other person did!! It's TOTALLY UNNECESSARY, unless you have no other means of factoring the trinomial. 

If you don't, I believe that you need to learn it!