Question

speed of river current is 6mph. ferry operator is looking to buy new boat . every day his route takes him 22.5 miles against the current and back to his dock and he needs to make this trip in a total of 9 hurs. he has a boat in mind, but he can only test it on lake where there is no current. how fast must the boat go on the lake in order for it to serve the ferry operator's needs?

Answer 1

Did we decide if it was a 22.5 mile roundtrip or a 45 mile round trip?

Please show your work.

Answer 2

I still don't understand what I am to do

Answer 3

Have you seen this? Distance = Rate * Time

Answer 4

yes

Answer 5

Perfect. That is all we need, other than a bunch of algebra and logic.

Rule #1 - Name Stuff.

Name what? What does the problem statement want?

"how fast must the boat go on the lake"

Okay, it wants the speed of the boan in still water. Perfect. Name that.

B = Speed of the Boat in Still Water.

This is an important abstraction. Are you okay with it, so far?

Answer 6

yes

Answer 7

This problem has some additional ambiguity that we must reconcile. The trip UP the river and the trip down the river TOTAL 9 hours. We do NOT know how long each trip took.

This is where we have to appeal to logic. If we are going AGAINST the river current, we will be going slower than when we are going WITH the river current. It should then be clear that the trip UP the river will take longer than the trip DOWN the river - given that both trips are the same length.

Still with me? This should make sense before we can move on.

Answer 8

yes

Answer 9

Sweet. Now, we need more naming. I'm going to ask you a dumb question in the hope that you will figure out how to answer it, even knowning that it is a dumb question. Don't worry about getting the answer wrong. Just give it your best guess.

How long is the trip UP river?

Answer 10

how many miles or the time

Answer 11

That was an answer I hadn't anticipated. Excelent work. Forcing me to be clear is very important. Let's try that again.

How long does it take to travel UP river as part of the round trip that takes 9 hours?

Answer 12

more than 4 1/2 hours

Answer 13

"More that 4½" is a very good answer. but that doesn't help us. We need to be EXACT. Can you do it? Think about the speed of the boat in still water that we defined above. How can we be EXACT with the time of the Up-River trip?

Answer 14

how do I know the time? All i know is it takes longer to go up than it does down and total up and down =9 hours

Answer 15

Perfect. That is the question I wanted you to ask. We simply don't know. Just like we didn;t know the speed of the boat in still water, we don't know the time of the up=river trip. If we don't know, let's just call it something and play like we know.

It's that abstraction thing, again.

U = Time it takes for the Up River Trip.

How's that? How long does it take for th up-river trip? U hours. Done.

Waht do you think? Still making sense?

Answer 16

yes

Answer 17

Okay, now this is a tough one. You have to get it. It's not impossible. There is enough inforamtion for a couple fo different and EXACT answers.

How must time is required for the down-river trip?

Answer 18

again I don't think I kknow the exact time but it would be less than 4 1/2 hours. should I know the exact time from info given?

Answer 19

That's okay. You'll get it.

How long does it take for the down-river trip? We don't know, so let's call it something and play like we know it! Works EVERY time. Get it stuck in your head.

How long does it take for the down-river trip? How about D hours? Is that a good name or is there something better we could call it? Keep in mind that the entire trip takes 9 hours.

Answer 20

D is great

Answer 21

It is great, but in this case, there is something a little better. Besides, we named

B = The Speed of the4 Boat in Still Water
U = The time it takes for the Up-River Trip

Now we have D = the time it takes for th down-river trip

How many different names are we going to have!!!!

Since we know the whole trip takes 9 hours, we can say U + D = 9. Agreed?

Answer 22

yes I just wrote that equation on my paper

Answer 23

Superb!!! Okay, then rather than calling the down-river trip "D", why don't we call it "9 - U"? How's that? Do you see where it came from?

Answer 24

yes

Answer 25

Awesome! We're all done with abstraction and logic. The rest is algebra!

Let's use Distance = Rate * Time. I'll write the up-river trip.

22.5 miles = (B - 6 mph)*(U hrs)

You write the down-river trip.

Answer 26

22.5 miles=(B+6)*(9-U)

Answer 27

Good. You should be more careful with your units, but it will do.

Now what?

Answer 28

what do u mean ? I should label? I don't know what to do next since I have 2 variables.

Answer 29

See how mine has hours and mph in it? That's what I mean. Making sure your units match WILL save you! You could blow up Jupiter when you are only trying to make pancakes, or something.

We have two equations and two variables. Remember what we did with "D"? Can we now do something similar with U?

Answer 30

9-D=U but that still leaves multiple variables. How does that help? What am i missing?

Answer 31

22.5 miles = (B - 6 mph)*(U hrs)

22.5 miles=(B+6 mph)*(9 hrs - U hrs)

Let's take a good, hard look at that first equation. What happens when we solve for U hrs?

\(U\;hrs = \dfrac{22.5\;miles}{B - 6\;mph}\) We have something else to call "U hrs"!!

Still with me?

Answer 32

i see what u did but don't know what else we can call u hrs.

Answer 33

We can call it what we just wrote on the other side of the equals sign. Let's rewrite the second equation with this new defintiion of "U hrs".

\(22.5\;miles=(B+6\;mph)*(9\;hrs - U\;hrs)\)

Becomes

\(22.5\;miles=(B+6\;mph)*(9\;hrs - \dfrac{22.5\;miles}{B - 6\;mph})\)

btw - You are a real trooper sticking with this. We're almost there!

Answer 34

ok i see that u substituted U with 22.5/B-6. now what

Answer 35

Careful, that shoudl be 22.5/(B-6).

We now have ONE equation in ONE unknown quantity!!!!!! We can solve for B!

What's first?

Answer 36

i give

Answer 37

After all that?! Doing a quick double-check, it look slike we got all the units right. Let's now simplfy our life and get rid of them, just for the sake of making hte algebraic manipulation simpler.

\(22.5 = (B+6)\cdot\left(9 - \dfrac{22.5}{B-6}\right)\)

Answer 38

i have that much but don't know how to solve for B

Answer 39

Lot's of ways to go about it...

Multiply by B-6

\(22.5\cdot(B-6) = (B+6)\cdot\left(9\cdot(B-6) - 22.5\right)\)

Answer 40

Sinplify the last parentheses

9B - 54 - 22.5 = 9B - 76.5

\(22.5\cdot(B-6) = (B+6)\cdot(9B-76.5)\)

How are we doing? What's next?

Answer 41

22.5B-135=

Answer 42

And the other side?

Answer 43

working on it

Answer 44

9B^2-21.5B-459=22.5B-135

Answer 45

right so far?

Answer 46

Gaa! So close. "-22.5B" on the left.

Answer 47

-22.5B-135? Why is 22.5 negative?

Answer 48

54 - 76.5

Answer 49

now what?

Answer 50

I think we might be confusing left and right. I meant the left in the last form that you wrote it.

9B^2 - 22.5B - 459 = 22.5B - 135

It was the 2 that was the problem, not the sign.

Okay, now what?

Answer 51

got it yes I was confusing sides

Answer 52

More simplifying! Combine like terms.

Answer 53

B2=5B+36

Answer 54

B=41???

Answer 55

Okay on the simplification, but \(B^{2} - 5B - 36 = 0\) might be a more familiar form.

Answer 56

yep

Answer 57

I'll just ignore that "41" thing. :-)

You have some factoring in your future!

Answer 58

give me some hints on factors for 36 that equl 5

Answer 59

-36 = -9 * 4 -- I'll just tell you, since we've been at this so long.

Answer 60

duh

Answer 61

I can do this all night, but I'm worried about your survival...

Answer 62

(B-9)(B+4)=0

Answer 63

Remember the other thread where I said we would be faced with 9 and -4?

Answer 64

are u a math teacher or do u just love doing math

Answer 65

i thought it is -9 and 4

Answer 66

Tough question. Math teacher? Not officially, but definitely at heart.

We have one more thing to do. Answer the original question! Remember what it was?

Answer 67

how fast must the boat travel in lake to work with current

Answer 68

B-9 = 0 leads to B = 9

B+4 = 0 leads to B = -4

As you pointed out before, B = -4 is a silly answer. Let's just throw it out.

Answer 69

We have determined that the trip will take EXACTLY 9 hours with a boat that goes EXACTLY 9 mph. Anything greater than 9 mph will also do. I'd be a little worried about exactly 9 mph. Anything at all goes wrong and you're done!

Answer 70

got it set each equation = to 0 and with 9mph in lake

Answer 71

You get the award for patience and fortitude.

For the record, a problem like this might take me about three minutes, just to write it all down. I am kind of a freak, so it might take humans a little longer than that. We just spent more than an hour on this one problem, but I hope we learned a LOT more than just how to solve this problem. We learned some algebra, how to name things when we get stuck, how to rename things, how to substitute, how to decode a compicated problem statement, how to solve a multi-variable problem, and frankly, how to get along!

Excellent work. Do some more and get faster at it!

Answer 72

thanks for sticking with me and explaining step by step

Answer 73

No worries. It's what we expect to do around here.