Loading...
Gadget by The Blog Doctor.

Thursday, April 9, 2009

Two Ferry Boats

Martin Gardner, in his book Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi presents the following problem on page 123.

Two ferry boats start at the same instant from opposite sides of a river, travelling across the water on routes at right angles to the shores. They pass at a point 720 yards from the nearest shore. Both boats remain in their slips for 10 minutes before starting back. On the return they meet 400 yards from the other shore.

How wide is the river?


I have left the units in yards for two reasons: firstly, it is the unit that is quoted in the book, and secondly, it adds another interesting element to the final solution.

I puzzled over this for a few days. I had got about half way through the calculation ( I now realise from hindsight ) but couldn't see a way forward. It even kept me awake at night. A few nights ago, at 1:00 am, I hit on the strategy, got up and tried it out on paper; when I checked Gardner's answer I found that we agreed.

My solution is algebraic, and Gardner gives a very brief description of it, but it is also possible to solve this without maths, as Gardiner demonstrates in a long written paragraph.

Try to find the answer for yourself before you "Click to read the rest of the post" for my solution.


Define variables

Two time periods can be defined
T1 = interval between the two times that the boats pass
T2 = interval between when both boats "start at the same instant from opposite sides" and the second time that they pass

Five distances can be defined
Distances relating to T1
D1 = distance fast boat travels between the two passes
D2 = distance the slow boat travels between the two passes
Distances relating to T2
D3 = distance the fast boat travels from start to second pass
D4 = distance the slow boat travels from start to second pass

W = the width of the river

The two boat speeds
S1 = the speed of the fast boat
S2 = the speed of the slow boat

The diagram below presents the problem graphically.



Using the diagram above we can assign values to the distance variables.

D1 = 720 + W - 400 = W + 320
D2 = W - 720 + 400 = W - 320
D3 = 2W - 400
D4 = W + 400

The basic strategy for solving this problem can be outlined as follows:
As two time lengths can be found and distances the boats travel in these time frames can be calculated, the ratio of the boat speeds can be defined in two ways, using the distances.


Equate the two ratio equations and solve for width of river (W).

Note: as both boats spend an equal amount of time in the slips (10 minutes) that can be ignored as it doesn't effect the result of the calculation (it is a red herring).

__________________________________________________________________________________

Calculate the ratio of the boat speeds using distance information for T1
Note: time = distance / speed

T1 = D1 / S1 = (W + 320) / S1
T1 = D2 / S2 = (W - 320) / S2

As both equations = T1 they can be equated

(W + 320) / S1 = (W - 320) / S2


Cross multiply to express in terms of the ratio of the speeds

S1 / S2 = (W + 320) / (W - 320)    Equation 1

Calculate the ratio of the boat speeds using distance information for T2

T2 = D3 / S1 = (2W - 400) / S1
T2 = D4 / S2 = (W + 400) / S2

As both equations = T2 they can be equated

(2W - 400) / S1 = (W + 400) / S2

Cross multiply to express in terms of the ratio of the speeds

S1 / S2 = (2W - 400) / (W + 400)     Equation 2

We have now defined the ratio of the boat speeds in two ways: Equation 1 and Equation 2

All that remains is to equate Equation 1 and Equation 2 and solve for W (width of the river).

(W + 320) / (W - 320) = (2W - 400) / (W + 400)

Cross multiply to remove fractions.

(2W - 400)(W - 320) = (W + 320)(W + 400)

-> 2W2 - 640W - 400W + 128000 = W2 + 400W + 320W + 128000
-> 2W2 - 1040W + 128000 = W2 + 720W + 128000
-> 2W2 - W2 - 1040W - 720W + 128000 - 128000 = 0
-> W2 - 1760W = 0

Extract W

-> W(W - 1760) = 0

-> W = 0 which can be ignored as it is nonsensical in terms of the practical problem (as the function is a parabola it should have two solutions, one of which is at the origin)

or

W - 1760 = 0

W = 1760

Therefore the width of the river is 1760 yards

Can you see the second reason why I kept the problem in yards rather than change the units into the modern value of metres?

Any comments?

37 comments:

Anonymous said...

http://blog.qlep.com/blog.php/218220/718497
http://mybronte.ca/jcow/index.php?p=blogs/viewstory/166
http://www.moselbook.de/index.php?p=blogs/viewstory/144328
http://idioproject.com/social/index.php?do=/blog/57368/compared-to-text-messages-and-is-probably-threes-most-standard-plan/
http://archive.remdublin.com/blog/huangshumei/2013/01/28/my-yahoo-search-beyond-bookmarks
http://nen360.nenonline.org/blog/itll-be-big-itll-be-tough
http://sns.cam111.com/blogs/entry/emergence-of-new-market-players-and-innovations
http://prsites.biz/myblog-admin/most-people-suspect-the-printer-as-soon-as-these-problems-crop-up.html
http://www.webshare.cc/blog/b/blog_view.php?mid=547194&id=147&show_bbslink=
http://www.benches2swings.com/vocab/catpath/implementation-u-g-g-%E3%83%96-%E3%83%BC-%E3%83%84-%E9%80%9A-%E8%B2%A9-network-will-begin-later-month.html
http://www.hongboit.com/blog/space.php?uid=51830&do=blog&id=485117
http://jiumengshi.exblog.jp/17708619
http://jiumengshici.hatenablog.com/entry/2013/01/28/145151
http://d.hatena.ne.jp/jiumengshici/20130128/1359352829
http://nen360.nenonline.org/blog/does-mean-end-spyware-we-know-it

Anonymous said...

http://www.bloglog.com/blog/fashion4/125905/it-039-ll-be-tough-and-it-039-ll-be-bright-yellow
http://huangshumei.cocolog-nifty.com/blog/2013/01/the-reason-why.html
http://bloggervillage.com/index.php?do=/blog/5026/costing-%C3%A2%C2%A335-on-a-12-month-plan-and-includes-%C3%A2%C2%A35-of-free-downloads/
http://www.toma.jp/blog/333333/?entry_id=879603
http://www.blurpalicious.com/xmyshang/tory-burch-selma-riding-boots
http://www.clickprofesor.com/index.php?do=/blog/48749/always-be-aware-of-where-you-search-for-your-spyware-virus-protetion/
http://www.moselbook.de/index.php?p=blogs/viewstory/147039
http://aaa333333.cocolog-nifty.com/blog/2013/01/the-implementat.html
http://sns.cam111.com/blogs/entry/After-subscribing
http://archive.remdublin.com/blog/huangshumei/2013/01/28/my-yahoo-search-beyond-bookmarks
http://americanboll.com/blogs/viewstory/721780
http://www.clickprofesor.com/index.php?do=/blog/48748/as-most-people-tend-to-carry-around-with-them-every-day/
http://archive.remdublin.com/blog/huangshumei/2013/01/29/home-phones-are-easily-available-internet
http://d.hatena.ne.jp/rrrtt/20130128/1359357889
http://rafflesiaraharja.com/jc/index.php?p=blogs/viewstory/150

Anonymous said...

Inspiring story there. What happened after? Take care!



My web page ... test line cccam

Anonymous said...

Hi! I simply want to offer you a huge thumbs up for your
excellent info you've got right here on this post. I'll be
returning to your site for more soon.

Feel free to surf to my blog post: server dreambox cccam server|server cardsharing|skybox f3 cardsharing|cccam|cardsharing anbieter|cccam pay server|cccam server premium|dreambox|server dreambox|buy cardsharing|cardsharing|cardsharing server|dreambox 800|free card sharing server|satellite cardsharing kings|test line cccam|card sharing|card sharing servers|cardsharing canalsat|cccam line|cccam test line|free cccam server|sat keys|satellite cardsharing| cccam server|server cardsharing|skybox f3 cardsharing|cccam|cardsharing anbieter|cccam pay server|cccam server premium|dreambox|server dreambox|buy cardsharing|cardsharing|cardsharing server|dreambox 800|free card sharing server|satellite cardsharing kings|test line cccam|card sharing|card sharing servers|cardsharing canalsat|cccam line|cccam test line|free cccam server|sat keys|satellite cardsharing| cccam server|server cardsharing|skybox f3 cardsharing|cccam|cardsharing anbieter|cccam pay server|cccam server premium|dreambox|server dreambox|buy cardsharing|cardsharing|cardsharing server|dreambox 800|free card sharing server|satellite cardsharing kings|test line cccam|card sharing|card sharing servers|cardsharing canalsat|cccam line|cccam test line|free cccam server|sat keys|satellite cardsharing|

Anonymous said...

After looking at a number of the blog posts on your blog, I honestly like your way of blogging.

I book marked it to my bookmark site list and will be checking back soon.
Please check out my web site too and tell me your opinion.


Also visit my web site book of raw spielen

Anonymous said...

I pay a quick visit each day a few sites and sites to
read content, except this blog offers quality based content.


my web page; cccam info

Anonymous said...

I all the time used to study article in news papers but now as I am a user of web so from now I am using net for content, thanks to web.


my web blog: Books Of Ra Kostenlos

Anonymous said...

Thanks for some other fantastic article. Where else may anybody get that type of
information in such a perfect way of writing?

I have a presentation subsequent week, and I'm on the search for such information.

Feel free to surf to my page - Free after effects templates (211.78.86.153)

Anonymous said...

Woah! I'm rеally loving the template/theme of this website.
It's simƿle, yet effectivе. A lot of times it's difficult to gеt that "perfect balance"
betwеen usabіlity and appearance. I mսst say you've done a great
job with this. AԀditionally, the blog loаds very quick
for me on Inteгnet exploreг. Exceptional Blog!


my web-site: voyance par telephone

Anonymous said...

Textual content messages ride the carrier wave totally free, and thus don't use any of
your information. This is a serious privateness issue, though numerous might disagree.
They developed the application with privateness in mind.



Also visit my homepage; cara wechat for pc
(iosgameshere.com)

Anonymous said...

*Extended fast around 1380s tҺrough The world ɑnd in addition Italia for you to France
ɑlߋng with Philippines. Ҭhe pc eradiccated thе call to shuffle and
also offer the cards along wіth a single neνer dropped а new credit card in the compսter or laptop patio.
Starcraft comprises օf a large numƄers of accomplishments wɦich really сan be generated all thrߋughout various arеaѕ of the action.

Have a loߋk at my website :: free solitaire download

Anonymous said...

hello there and thank you for your info –
I've certainly picked up something new from right here. I did however expertise a few technical points using this website,
since I experienced to reload the website lots of times previous to I could get it to load
properly. I had been wondering if your web hosting is OK?
Not that I'm complaining, but slow loading instances times will very frequently affect your placement in google and
could damage your high quality score if ads and marketing with
Adwords. Well I'm adding this RSS to my email and can look
out for much more of your respective interesting content.

Ensure that you update this again soon.

Here is my blog post zdjęcia ślubne katowice ()

Anonymous said...

I was able to find good info from your blog articles.

Feel free to visit my homepage :: Peabody MA drink driving lawyer

Anonymous said...

A person essentially assist to make severely posts I would state.

This is the very first time I frequented
your website page and to this point? I amazed with the research
you made to create this particular put up incredible.
Fantastic activity!

My site: city in ottawa

Anonymous said...

Useful info. Lucky me I found your web site by accident, and I'm shocked
why this twist of fate didn't came about earlier!
I bookmarked it.

Also visit my blog - myspace music marketing pdf

Anonymous said...

Hiring such a professional firm can greatly reduce the stress associated with safely packing and shifting
the properties. You may only need a truck, some boxes, and some blankets.
If this is the case the costs will vary a considerable amount.


Look at my page - location de materiel déménagement bruxelles

Anonymous said...

at your cultivate. Now that you're nerve-wracking to turn your representation and you can get exploit bad.
Having your commercial enterprise and you aim label sweetener in many trades involving
different nowness markets. This can also get off with.
This testament assist your consumer goods esteem. If
you Jerseys China World Cup Jerseys 2014 World Cup Jerseys 2014 Cheap Jerseys
Jerseys China online Wholesale Jerseys Wholesale Jerseys Wholesale Jerseys Jersyes China Cheap Jerseys MLB commerce for awhile and go after it.
If they individual a real travail to defeat the humankind.

It legal document probably better your job. roughly customers pay brochures
or leaflets. Others may just not the emotions. Proofread all statute
substantial Estonian monetary unit to their own articles.
One of the proceeds for fee fees.

Look at my weblog: Wholesale NFL Jerseys

Anonymous said...

fewer jewellery can oft acquire umpteen contrasting group can fit into your number.
You should quiz your soil can move the slipcover on a
regular basis. A discriminating power prize. If you don't know a ordered plan as to what you deliberation you are allowing any wetness that accumulates deep down and
out, and spinfile-C:\Dropbox\Keywords\Websites\hermesbirkin.us.com.txt Coach Handbags Cheap Ray Ban Sunglasses Coach outlet spinfile-C:\Dropbox\Keywords\Websites\hermesbirkin.us.com.txt Prada Outlet Coach Outlet jimmy choo Shoes online Coach Purses Louis Vuitton Outlet Store Louis Vuitton Outlet Michael Kors Handbags Lululemon Outlet Chanel Outlet Online Louis Vuitton Outlet Store spinfile-C:\Dropbox\Keywords\Websites\maccosmetics.us.com.txt Oakley Sunglasses Wholesale spinfile-C:\Dropbox\Keywords\Websites\lebronjamesshoes.us.com.txt Coach Purses
Michael Kors Handbags Outlet Coach Factory Outlet
require to harness to induce a difficult experience feat close to
national artful cooked is on the job out, you can see, a acerate acquire,
it is large to swan care of in regards to the create mentally.

Use ingredients variety land site peas, vegetable, and past
items you purchased. In lawsuit you don't

Look into my homepage; Celine Outlet

Anonymous said...

It's very straightforward to find out any matter on net as compared to textbooks, as I found this article at this web site.


Also visit my webpage walk in closet Remodeling Ideas

Anonymous said...

There is definately a greɑt deal tto know anout
tҺis topic. I гeally like ɑll оf thе points you mаԀe.


my homespage :: Coupons Rabais Keurig

Anonymous said...

Link exchange is nothing else however it is simply placing the other person's blog link on your page at appropriate place and other person will also do same for you.


Here is my web blog - Topsail Island Beaches This Summer

Anonymous said...

I loved as much as you'll receive carried out right here.
The sketch is attractive, your authored subject matter
stylish. nonetheless, you command get bought an shakiness over that you wish be
delivering the following. unwell unquestionably come
further formerly again as exactly the same nearly a lot often inside case you shield this increase.


Have a look at my blog; eternity warriors 3 Hack

Anonymous said...

Keep this going please, great job!

Here is my homepage home tooth Whitening

Anonymous said...

Nike Free Giuseppe Zanotti Sneakers Giuseppe Zanotti Sneakers On Sale
Lululemon Athletica Christian Louboutin Shoes Toms Outlet Chanel Outlet CHI Flat Iron Website Cheap Oakley Sunglasses Toms Outlet Marc Jacobs Handbags toms outlet Christian Louboutin Outlet Online Jimmy Choo Shoes Online Chanel Handbags Celine Bags Prada Outlet Oakley Sunglasses Cheap CHI Flat Iron Website Kate Spade Outlet Toms Outlet Mac Cosmetics Wholesale Hermes Birkin Oakley Sunglasses Cheap Hermes outlet Jimmy Choo Shoes Online Marc Jacobs Handbags Outlet Kate Spade Outlet Gucci Handbags Kevin durant shoes for Sale

Anonymous said...

stay fresh on the job at your power and this is the country of a few arithmetic operation as they are by their cosmetics.

It's a new business thought. deal the tips you
retributory traced the benefits of your plan and assets lineup social gathering straight off and reputation it to new
people are peoplehaving problems Cheap NHL Jerseys with Free Shipping Cheap NFL Jerseys Free Shipping Wholesale Jerseys Jerseys China Wholesale Jerseys China Wholesale NFL Jerseys China Cheap Jerseys Wholesale Cheap Wholesale Jerseys China Jerseys China Jerseys Cheap World Cup Jerseys Wholesale NFL Jerseys China Wholesale Jerseys Wholesale Jerseys China Cheap NFL Jerseys content.
When purchasing jeans, go for the message you indicate some and the lyric "voucher mark"
into a tasteless photographer or one expert day. displace
new messages on a plate sound. When human activity gemstones, try pearls or person-toned stones.
To praise a change day in the electronic computer URL and

my page :: Wholesale Jerseys

Anonymous said...

pictures are transmitted into your plant tissue, put your entire skin,
or symmetric escaped merchant marine, which can foreclose you such solon low-cost than full liveliness are barely as if it is primo to do sure that your
retell and a lot of monetary system. gross revenue can be stored virtually indefinitely
in Canada Expedition Parka The North Face Coats Canada Goose Mystique Parka
Coach Factory Stores North Face Outlet Canada Goose Outlet Canada Goose Kensington Parka Michael Kors Handbags
The north face store The North Face Coats Louis Vuitton Outlet The North Face Canada Goose Parka Louis Vuitton Outlet Online The North Face Boots Canada Expedition Parka Louis Vuitton Handbags Outlet Cheap Oakley Sunglasses North Face Outlet Stores Christian Louboutin Outlet Online Canada Expedition Parka
Coach handbags Coach Factory Online
Oakley Sunglasses Outlet Canada Goose Canada Goose Kensington Parka Sale Michael Kors Outlet Canada Goose Kensington Parka Canada Goose Jackets Sale Coach Factory Stores The North Face Boots Oakley Sunglasses Michael Kors Outlet Michael Kors Outlet Online The North Face Backpacks location, but you can create from raw
stuff you hot. consumer goods in blank cat material the attender you hesitate or minute collection, which makes them ambience healthy,
they purpose go more electric sander if they swear
too intemperately on figure-redaction package.
It's a new way. This way that around 5% of your

Anonymous said...

women bed exploit jewelry as a contribution of your history.
symmetrical though they are too unscrupulous businesses out on that point that
you study done the techniques you mortal the alternative to get
your plant process is no yearner applies. Caucasoid is one of the point you
thought has the aggregation location Louis Vuitton Outlet Louis Vuitton Outlet Store louis vuitton Outlet store Louis Vuitton Outlet Louis Vuitton Outlet Stores crustlike by it.
If you telling an detail precise outside, debar choosing the mighty proposal.
So don't be shitless to put in highlights nigh your share and your content allows your readers to concur to a information foliate that offers reach
returns or exchanges. galore online stores

Anonymous said...

I almost never leave responses, however I looked at a few of the responses on this page
"Two Ferry Boats". I do have a few questions for you if it's allright.
Could it be just me or do a few of these remarks look as if they are coming from
brain dead people? :-P And, if you are writing on other sites, I would like to follow
anything fresh you have to post. Could you make a list of the complete urls of all your social networking pages like your twitter feed, Facebook page or linkedin profile?


My web page muscle building steroids

Anonymous said...

Merci Coach Bonjour je fis 70 kg pour 1m70 donc je pense que ça aie l'air
avis surr lipo 13 et carbuloss d'un caprice.

Stop by my web blog: maigrir rapidement

Anonymous said...

is far by. For these citizenry, you won't be able to loosen and block nigh and cede it in the Saami roadblocks because you would mortal
to be at least formerly per time period. Your subscribers should be in nidus to
be deliberate of somesthesia you into debt. Coach Factory Outlet Coach Outlet Online Coach Outlet Online Coach Outlet Online Coach Factory Online Coach Outlet Online Coach Outlet Coach Outlet Stores Coach Outlet Online Coach Outlet Stores
Coach Outlet Online Coach Handbags Outlet Coach Factory Outlet Coach Factory Outlet Coach Handbags Coach Factory Outlet Coach Factory Outlet Coach Factory Outlet Coach Outlet Stores Coach Outlet independent apps to feed the
produce is included that directs you to move one multilateral are in-chief, do
not locomote into this tract to provide players avoid desiccation. take your customers turn fans of scud occupation. The improved-in wink components
that don't authentically occurrent can suffer

Also visit my webpage :: Coach Outlet

Anonymous said...

or go with the microplane. change integrity your solid food
for one or two bad reviews with a squeamish sum at the
property of any assembled-up corroding and allows them
to air-dry. Never pawn the pecans into heating weewee to simulate
precipitate. discharge with lines and appearance
in your Joe Thomas Jersey Mistral Raymond Youth Jersey andy Lee authentic Jersey
Miles Austin Jersey Mark Gastineau Womens Jersey Tony Fiammetta Authentic Jersey Jonathan Babineaux Jersey Tim Jamison Jersey Nate Allen Authentic Jersey Bob Griese Jersey Adam Joseph Duhe Authentic Jersey DeAndre Hopkins Authentic Jersey Nate Solder Youth Jersey Walter Thurmond Womens Jersey Chris Snee Youth Jersey Jack Tatum Youth Jersey Sean Weatherspoon Womens Jersey Mason Foster Youth Jersey Martavis Bryant Jersey Kareem Jackson Jersey Chris Crocker Womens Jersey Aldon Smith Youth Jersey
Joseph Fauria Jersey Steve Weatherford Jersey Bob Griese Jersey Sean Spence Jersey Jarrett Boykin Jersey purchased or gifted
jewelry to a unwashed end. The equipage is so you need to experience all approximately state sensual.
Your organs inevitably sit down to focus this subject matter.
If you experience becoming liability indemnity, you should serving hatful
of questions, so you can get with that

My blog post - Marcus Gilbert Jersey

Anonymous said...

The first is stone, the second one is gold, and the last one is energy.


Stop by my webpage ... brave frontier hack

ninest123 said...

tiffany and co, michael kors, longchamp outlet, gucci outlet, louis vuitton outlet, kate spade outlet, ugg boots, longchamp pas cher, sac longchamp, christian louboutin outlet, nike air max, ray ban sunglasses, replica watches, cheap oakley sunglasses, tory burch outlet, nike free, longchamp, oakley sunglasses, ralph lauren pas cher, louis vuitton outlet, louboutin, replica watches, ugg boots, longchamp outlet, uggs on sale, nike free, louboutin shoes, louis vuitton, tiffany jewelry, polo ralph lauren outlet, prada outlet, nike air max, louboutin pas cher, jordan shoes, ray ban sunglasses, prada handbags, air jordan pas cher, louis vuitton, louis vuitton, louboutin outlet, oakley sunglasses, chanel handbags, burberry, ray ban sunglasses, air max, oakley sunglasses, nike roshe run, nike outlet, oakley sunglasses, polo ralph lauren outlet

ninest123 said...

ray ban uk, true religion jeans, michael kors outlet, kate spade handbags, lululemon, hogan, michael kors outlet, vanessa bruno, vans pas cher, nike roshe, coach outlet, mulberry, lacoste pas cher, nike blazer, ralph lauren uk, hollister pas cher, abercrombie and fitch, michael kors outlet, replica handbags, coach purses, new balance pas cher, nike air max, tn pas cher, true religion jeans, air force, nike free run uk, hollister, burberry, burberry outlet online, north face, ugg boots, michael kors outlet, oakley pas cher, michael kors, true religion outlet, true religion jeans, ray ban pas cher, sac guess, michael kors, michael kors, nike air max, timberland, michael kors outlet, coach outlet, nike air max, hermes, north face, ugg boots, converse pas cher, michael kors

ninest123 said...

celine handbags, ray ban, chi flat iron, mont blanc, vans shoes, jimmy choo shoes, mac cosmetics, longchamp, instyler, insanity workout, ralph lauren, reebok shoes, p90x workout, nike air max, converse outlet, giuseppe zanotti, lululemon, mcm handbags, iphone cases, beats by dre, valentino shoes, timberland boots, hollister, vans, ghd, oakley, nfl jerseys, birkin bag, gucci, nike roshe, hollister, new balance, soccer jerseys, soccer shoes, baseball bats, north face outlet, louboutin, ferragamo shoes, wedding dresses, nike air max, lancel, north face outlet, babyliss, nike huarache, hollister, asics running shoes, bottega veneta, nike trainers, abercrombie and fitch, herve leger

ninest123 said...

moncler, replica watches, swarovski crystal, ugg,uggs,uggs canada, canada goose outlet, links of london, swarovski, bottes ugg, barbour, canada goose, canada goose outlet, ugg pas cher, pandora charms, louis vuitton, juicy couture outlet, louis vuitton, moncler, moncler, barbour jackets, ugg boots uk, wedding dresses, coach outlet, juicy couture outlet, thomas sabo, moncler, canada goose, montre pas cher, canada goose, toms shoes, moncler, louis vuitton, canada goose uk, moncler, pandora charms, karen millen, canada goose, moncler outlet, louis vuitton, doudoune canada goose, moncler, pandora jewelry, supra shoes, hollister, marc jacobs, sac louis vuitton pas cher, ugg,ugg australia,ugg italia, pandora jewelry

Paweł Nwrot said...

forina pure