Question

I really need help! MEDAL!!!!
The speed of the current in a stream is 5 mi/hr. It takes a canoeist 300 minutes longer to paddle 5.5 miles upstream than to paddle the same distance downstream. What is the canoeist's rate in still water

Answer 1

you know about vectors?

Answer 2

canoe + stream is faster, but the same distance
canoe - stream is slower, but the same distance

since the distance remains the same, they have to equal each other out

what is a good formula to find the distance covered if we know speed and time?

Answer 3

No
These are makeup assignments. So I'll need explanation

Answer 4

And I honestly have no idea.

Answer 5

this is a key part of the solution, youll need to find it before we move on. otherwise ill just feel like im doing all the work/thinking.

we need to know how to find distance, using speed and time.

Answer 6

if im running at 3 feet per sec, and i run for 5 sec ..... how far have i gone?

Answer 7

distance=speed/time?

Answer 8

soo close

speed is measured in distance per time

s = d/t, therefore d = st

Answer 9

ohh, okay sorry

Answer 10

3 feet/sec * 5sec = 15feet*sec/sec = 15feet

Answer 11

anywhos

we know the distance is the same sooo

speed(up) * time(up) = speed(down) * time(down)

this make sense?

Answer 12

yes

Answer 13

so it basically balances out

Answer 14

yep

up stream fights the current, so speed(up) = (canoe - river)
down stream goes with the current, so speed(down) = (canoe + river)


what do we know about the time frames?

Answer 15

Well, the time frames are different? If that's a start

Answer 16

should we subtract them?

Answer 17

going 5.5 downstream takes 72 minutes going upstream its 372 minutes

Answer 18

they are different :)

takes 300 minutes longer to paddle upstream than downstream.

time(up) = time(down) + 300

Answer 19

soo plugging all this in:

(canoe-river) (time(down)+300) = (canoe + river)(time(down))

river = 5

(c-5) (t+300) = (c+5)(t)

any ideas from this?

Answer 20

cancel the c-5 and the c+5?

Answer 21

they dont cancel because they are not equal.

and i just wrote up something that i couldnt verify so i deleted it.

the distance traveled plays into this

Answer 22

oh okay
could you divide by t?

Answer 23

and solve to get c?

Answer 24

ugh, this isnt panning out for me, im missing something obvious :/

im thinking ... i need elimination for this?

(c-5) (t+300) = d
(c+5)(t) = d


c(t+300)-5(t+300) = d
ct +5t = d


ct+300c-5t-1500 = d
ct +5t = d

ct+300c-5t = d+1500
ct +5t = d

now we add
-------------------------

ct+300c-5t = d+1500
ct +5t = d
--------------------
2ct+300c = 2d + 1500


not it

Answer 25

im taking this to paper, typing is a bother

Answer 26

I'm getting confused

Answer 27

lol

Answer 28

i was seeing if my thought was getting somewhere, but its just running in circles :) let me organize it and see if we can approach this from the right angle is all

Answer 29

okay, sounds great! thank you for helping by the way!

Answer 30

speed = d/t

c - r = d/(t+300)
c + r = d/t

elimination is the process where we remove one of the variables from the playing field; if we multiply the top equation by -1, and add the results we remove c.

-c + r = -d/(t+300)
c + r = d/t
-----------------
2r = d/t - d/(t+300)

r and d are known so we can find t, right?

Answer 31

yes

Answer 32

r = 5mi/60min
d= 5.5

2(5/60) = 5.5(1/t - 1/(t+300))

can you tell me what t is?

Answer 33

(5.5/t)-(5.5/t+300)????

Answer 34

2(5/60) = 5.5(1/t - 1/(t+300))

5/30 = 5.5( (t+300-t)/t(t+300) )

1/6 = 5.5(300)/(t(t+300))

t(t+300) = 5.5(6)(300)

t(t+300) = 5.5(6)(300)

t^2 + 300t = 9900, which is a quadratic equation, not sure if youve covered these.

Answer 35

We should go through it just for learnings sake

Answer 36

maybe ....

the shortcut is the quadratice formula

given: ax^2 + bx + c = 0

x = [-b +- sqrt(b^2-4ac)]/(2a)

in our case

\[t=\frac{-300+\sqrt{300^2+4(9900)}}{2}\]

or t=30

Answer 37

so time is 30?

Answer 38

time down stream is 30 minutes if ive kept my wits about me :)

Answer 39

so,

c+r = d/t, we know it all but c

c = d/t - r

c = 5.5/30 - 5/60

Answer 40

c=6/60?

Answer 41

now that sounds better, yes

6 miles per 60 min = 6 miles per hour

Answer 42

so the answer is 6?

Answer 43

with any luck, yes :)

Answer 44

I'll check it really quick and tell you!

Answer 45

and im sure ive taken a long way to get there, since the answer to this tends to be just a few short rows ...

Answer 46

It's right!

Answer 47

thank you for your help!

Answer 48

youre welcome, if i can come up with the shorter solution process, ill post it lol

Answer 49

haha okay, thanks again!

Answer 50

using my first thought:
\[(c-r)(t+300)=(c+r)(t)\]
\[\frac{t+300}{t}=\frac{c+r}{c-r}\]
\[1+\frac{300}{t}=\frac{c+r-r+r}{c-r}\]
\[1+\frac{300}{t}=1+\frac{2r}{c-r}\]
\[\frac{300}{t}=\frac{2r}{c-r}\]
\[\frac{300}{t}=\frac{k(2r)}{k(c-r)}\]
\[300=k\frac{2(5)}{60}=k/6\\
k=1800\\
t=1800(c-r) \]
\[c+r=\frac{d}{1800(c-r)}\]
\[c^2-r^2=\frac{d}{1800}\]
\[c=\sqrt{r^2+\frac{d}{1800}}\]
\[c=\sqrt{(5/60)^2+\frac{5.5}{1800}}=1/10\]
1 mile per 10 minutes, so 6 miles per 60 minutes

6 miles per hour

Answer 51

it aint shorter, but at least we aint got to go all quadratic on it :) good luck