Distance, rate, and time

Date: 16 Nov 1994 14:36:39 -0800
From: Brianne Ashby
Organization: NWT School BBS
Subject: Dolphin swimming - Dist/Rate/Time

There is a dolphin swimming at 250 km per hour and the water flows 
at 150 km per hour and he is trying to get to his destination, 
which is 500 miles away. How long would it take him to get to his 
destination?

Bye!
             From Brianne and Melissa

Date: Wed, 16 Nov 1994 22:33:33 -0500 (EST)
From: Dr. Morton
Subject: Dolphin swimming - Dist/Rate/Time

Hello Brianne and Melissa!

I'm Mike, one of the Math doctors here to answer your question...
so, let's take a look at it  :)

Let me first introduce an equation you may or may not be familiar 
with:

                Distance = rate x time
(rate is speed)
                       d = r x t

Can you see why? If we're in a car going 60mph for ONE hour, we've 
gone 60 miles because:

                        d = 60mph x 1 hour
                          = 60 miles

This equation can also be written as:

                        t = d/r

I'll leave it to you if you can see why  :)

So, let's draw a little picture:

   >
>ooo>ooo-   ------->    <----     ~~~~~~
Dolphin     250 km/h    150 km/h  Water

I'm assuming that the water is flowing OPPOSITE the direction the 
dolphin is swimming; AGAINST the dolphin.  If it's the other way, 
you should be able to figure out the problem almost the same as 
this way  :)

So, if the dolphin is going to the RIGHT at 250 km/h and the water 
is going to the LEFT at 150 km/h, then we must SUBTRACT them.  Can 
you see why this is true?  Since the dolphin is going AGAINST the 
water, he actually goes SLOWER because he must fight the current.  
So:
                        250 km/h - 150 km/h = 100 km/h

The dolphin is actually moving 100 km/h!

Therefore, using our first equation (and converting 500 miles to
kilometers, getting 500 x 1.6 km = 800 km), we have

        time = distance / rate
             = 800 km / 100 km/h
             = 8 hours!

I hope that makes everything clear.  Let us know if you don't 
understand or if you need anything else :)

- Morton, doctor of sorts   The Math Forum
  Check out our web site:  http://mathforum.org/dr.math/   

Date: Mon, 16 Dec 1996 18:17:02 +0100
From: ASL Webmaster
Subject: Dolphin swimming - Dist/Rate/Time


My math tutor and I popped in to your site for some math word 
problems. After puzzling through the problem about the dolphin, we 
thought we'd better add our two cents.  

First, Morton (doctor of sorts -- hey, I have a friend who calls 
himself "Doc", too, but makes no claim to professorship) should 
have mentioned a few things:

1. Dolphins are streamlined and aren't completely affected by the
   current.  Just how affected they are would require some extra  
   experimenting and data collecting.

2. We don't know which way the current is moving.  If it's coming
   on an angle, we have to know exactly what angle and my tutor 
   tells me that something called trigonometry - adding vectors 
   and so on - would have to be used to solve this which puts it 
   into the HS realm of math.

3. The problem states the distance as 500 MILES, so converting  
   into Km. would have to be done first.  Naughty naughty Dr. 
   Morton!

Anyway, we still love your word problems.  Keep 'em flying!  But 
maybe Dr. Morton could use a tutor, too!.

Your pals, Matt and Ted, docto of sorts.

From: Dr. Ken
Date: December 17, 1996
Subject:  Dolphin swimming - Dist/Rate/Time

Hi!

Let me address your points - you're absolutely right that Mike 
should have converted the miles to kilometers before solving this 
problem.  That was a slip-up, and it's now corrected.  Thanks for 
catching it.

In your second point, you're right that we aren't told which way 
the current is flowing, but Mike knew that and he said that he'd 
just assume that the current flows in the opposite direction that 
the dolphin swims.  Otherwise, without making some kind of 
assumption, we can't solve the problem.  If the current isn't 
flowing in the same direction or the opposite direction, then 
things get more interesting, and your tutor's right: we'd use 
vectors to solve the problem.  But we'd probably be able to solve 
it without using trigonometry (darn, because trig is pretty neat), 
again depending on how the problem is stated.  For instance, if I 
say that the dolphin swims East at 250 kph, and the water flows 
Northwest at 150 kph, then we don't need trig, but if I say that 
the water flows in the direction 18 degrees north of East, then 
you'd need trig.  It looks like a mess, but it's not too bad.

And now, what I think is your most interesting point: how the 
streamlining of the dolphin comes into the situation.  For this, 
I'll ask you to perform a little thought experiment.  Imagine a 
dolphin hovering underwater in the middle of the ocean, and 
imagine a big sheet of plastic hovering right next to the dolphin.  
Along comes a big 150 kph current which hits both the dolphin and 
the plastic sheet.  It flows straight into the nose of the dolphin 
(that is, the dolphin points straight into the current) and hits 
the plastic sheet face-on.  The sheet is a whole lot less 
streamlined than the dolphin, so observing their respective 
effects should give us insight into what streamlining means in 
this context.

Well, I say that the only effect streamlining has is that it will 
determine how long it takes for the floating body to speed up to 
150 kph.  The plastic sheet, having a whole lot of resistance and 
not much mass, will speeed up very quickly to 150 kph, whereas the 
more streamlined and massive dolphin will take a little while 
longer, but it will still get sped up all the way to 150 kph.  My 
feeling is that this will only take a few seconds, but this may 
take some experimentation.

So when you say that the dolphin isn't completely affected by the 
current, I'd have to say that streamlining just makes it take a 
little while longer for the dolphin to feel the effects, and then 
we could do our good old vector math on it.

A little side observation: this fluid resistance stuff happens all 
over the place - it's why your the shape of your eardrum is 
similar to that of a plastic sheet, and isn't shaped like a 
dolphin.  It's why a bird _is_ shaped like a dolphin when it 
flies.

- Dr. Ken   The Math Forum
  Check out our web site:  http://mathforum.org/dr.math/