SOLUTION: The rate at which a river flows is one-third the speed of a boat in still water. If that boat travels down that river for two hours and then back upriver for two hours, it will be
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Question 1145608: The rate at which a river flows is one-third the speed of a boat in still water. If that boat travels down that river for two hours and then back upriver for two hours, it will be 16 miles short of its starting point.
What is the speed of the boat in still river?
Please help--thanks
Here where I got so far-but the answer does not come out to the choices given
1/3 ÷ ( x + 2) + 1/3 ÷ (x - 2) =
Here are the choice of answers given: 4mph, 8mph, 12mph, 24mph
Found 3 solutions by VFBundy, Theo, ikleyn:
Answer by VFBundy(438) (Show Source): You can put this solution on YOUR website!
Speed of boat in still water = b
Speed of current = c
c = (1/3)b
Down the river:
d = d
r = b + c = b + (1/3)b = (4/3)b
t = 2
r = d/t
(4/3)b = d/2
b = (3/8)d
Up the river:
d = d - 16
r = b - c = b - (1/3)b = (2/3)b
t = 2
r = d/t
(2/3)b = (d - 16)/2
b = 3(d - 16)/4
b = (3d - 48)/4
b = (3/4)d - 12
b(down the river) = b(up the river)
(3/8)d = (3/4)d - 12
-(3/8)d = -12
(3/8)d = 12
d = 32
From earlier:
b = (3/4)d - 12
b = (3/4)(32) - 12
b = 24 - 12
b = 12
Speed of boat in still water = b = 12 mph
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
basic formula is r * t = d
r is the rate
t is the time
d is the distance.
when t = 2, the formula becomes 2 * r = d
let b = the rate of the boat.
let w = the rate of the water.
with the current, the formula becomes 2 * (b + w) = d
against the current, the formula becomes 2 * (b - w) = d - 16
this conforms to the requirement of the problem that that the return trip comes 16 miles short of the distance traveled when going with the current.
simplify these two equations to get:
2b + 2w = d
2b - 2w = d - 16
subtract the second equation from the first to get:
4w = 16
solve for w to get w = 4
you are given that the rate of the water is 1/3 the rate of the boat.
the equation for that is 1/3 * b = w
since w = 4, this becomes:
1/3 * b = 4
solve for b to get:
b = 12
you now have the rate of the boat is 12 mph and the rate of the water is 4 mph.
the rate of the boat is 12 mph is your answer.
you can confirm by replacing b and w in the original equations to get:
2b + 2w = d becomes 24 + 8 = d which gets you d = 32.
2b - 2w = d - 16 becomes 24 - 8 = d - 16 which gets you 16 = d - 16
solve for d to get d = 32.
replacing all variables with their values in the original equations gets you.
2b + 2w = d becomes 24 + 8 = 32 which becomes 32 = 32, which is true.
2b - 2w = d - 16 becomes 24 - 8 = 32 - 16 which becomes 16 = 16, which is true.
the values of b and w and d are confirmed to be true.
your solution is that the speed of the boat is 12 mph.
Answer by ikleyn(52782) (Show Source): You can put this solution on YOUR website!
.
How you start solving the problem in your post, clearly shows that you are unfamiliar with the subject.
It is a typical and standard Upstream and Downstream trip word problem.
You can find many similar fully solved problems on upstream and downstream trips with detailed solutions in lessons
- Wind and Current problems
- More problems on upstream and downstream round trips
- Wind and Current problems solvable by quadratic equations
- Unpowered raft floating downstream along a river
- Selected problems from the archive on the boat floating Upstream and Downstream
in this site, where you will find other similar solved problems with detailed explanations.
Read them attentively and learn how to solve this type of problems once and for all.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this textbook under the section "Word problems", the topic "Travel and Distance problems".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
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