## Mr Plow – Something for the British!

Posted by Karl Sharman on December 18, 2010 – 12:30 pm

Snow starts falling at some time before noon on a cold December morning. At noon a snow plow starts plowing a road (as if!!).

At 1 pm the plow has advanced 2 miles, but it takes another 3 hours to do the next 2 miles.

What time did it start to snow?

And is plow spelt plow or plough? Or both.

December 18th, 2010 at 1:07 pm

Assuming that the snow falls at a regular rate and that the plow’s speed is proportional to the volume of snow it has to go through, the snow would have started to fall an infinitesimally short time before noon.

(Spelt plough here in Oz where it rarely snows).

December 18th, 2010 at 2:56 pm

Hi Karl. There isn’t enough info. Has Wiz got the right speed proportional to the snow’s volume (depth presumably) relationship?

December 18th, 2010 at 3:08 pm

Not my question… but I have seen the answer, and there is enough information!

Snowfall is constant. There is a relationship between the speed of the plough (I prefer plough – it looks right) and the snowfall….

December 18th, 2010 at 4:04 pm

Hi Karl. Plough’s good for the UK. Plow’s good for Homer Simpson.

I guess that speed is inversely proportional to the snow depth, but that cannot be deduced from the information given. I wouldn’t expect that a real snow plough would be affected to much by thhe depth.

I accidentally said the the wrong way in my last post.

December 18th, 2010 at 5:08 pm

Let 12 noon be time = 0. The snow starts falling at time t = -t0

at constant rate d’, where d is the depth, and d’ is the rate of change of

depth wrt time. So at time t the depth will be d = d’(t+t0).

Assuming the speed of the plough, v = k/d.

Then the distance the plough travels, s = integral [T= 0 to t, k/(T+t0)]

s = k ln(t+t0) – ln( t0) = k ln((1 + t/t0).

When t = 1, s = 2, when t = 4, s =4, o

2 = k ln(1+1/t0)) and 4 = k ln(1+4/t0))

Combining => 2 ln(1 + 1/t0) = ln(1 + 4/t0)

=> (1+ 1/t0)^2 = (1 + 4/t0)

=> 1 + 2/t0 + 1/t0^2 = 1 + 4/t0

=> 1/t0^2 = 2/t0

=> t0 = 2 (as only physically meaningful answer).

The snow started at 10 am (barring silly mistakes).

December 18th, 2010 at 5:11 pm

Ooops. I made a minor slip, my k unintentially included d’ immediately after defining it, but it makes no difference to the 10 am result.

December 18th, 2010 at 5:13 pm

Aaargh. t0 = 1/2, so 11:30 am.

December 18th, 2010 at 5:32 pm

Simpsons – Mr Plow – I knew there was a reason why Plow stuck in my mind!!!

Where’s the Plow King or Mr Plow when I need them?

December 19th, 2010 at 7:15 am

Chris got there first and wins the contract for clearing snow in the UK!

Here is my solution:-

At time t, the amount of snow is some constant(r) times t. snow = rt.

The velocity at which the snow plough can travel is inversely proportional to the amount of snow: v = c/rt Since the plough travelled the same distance from noon to 1 as it did from 1 to 4, we can set these distances equal, and use the integral of the velocity as the distance.

The snow started at some time x before the plough, so the integrals are from x to 1+x, and from 1+x to 4+x. These integrate to: r/c(ln(1+x) – ln(x)) = r/c(ln(4+x) – ln(1+x))

The constants cancel, and rearranging use logarithmic properties gives: ln((4+x)(x)/(1+x)^2) = 0 (4+x)(x)/(1+x)^2 = 1 (4+x)(x) = (1+x)^2 Which simplifies to x=1/2.

So, the snow started at 11:30 AM.

December 19th, 2010 at 7:20 am

So, if there was no snow, the snow plough would be able to travel infinitely fast

December 19th, 2010 at 7:32 am

That is correct, Chris. And when he switches on his lights the beam of light will be traveling at warp 2…..

Breaking the speed of light barrier was easy…. what’s next?

December 19th, 2010 at 7:47 am

… won’t the light be travelling backwards

December 19th, 2010 at 10:40 am

Ah yes, infinity is faster than the speed of light – I=C^I

He will at least be able to see what he has done. But if he switches on the lights pointing backwards……

More garbage will be spouted later…

December 19th, 2010 at 2:56 pm

A straight line graph of time (x axis) and depth of snow (y axis) has the volume of snow accumulated as the area under the line. It is agreed that the plough’s speed is directly related to the volume of snow.

If snowfall = 0 when t = 0 then the area under the line between t = 2 and t = 4 is three times that between t = 0 and t = 2.

So snow started to fall at t = 0, i.e. noon, not 11:30 am.

December 19th, 2010 at 3:05 pm

Disregard the last post – mixed up time and distance.

D’oh!

December 19th, 2010 at 9:36 pm

I’m sorry to come late to the party, especially since the correct answer has already been given by both Chris and Karl. However, I do have two comments.

The first is that I once again have a problem with the way the problem was presented. There is simply not enough information given to make a really meaningful question. What I mean, is that we are not told whether or not the snowfall is constant, nor are we told whether or not the snow plow removes snow at a constant rate or not. If either or both of those things is variable, then we, of course will get completely different answers. Chris and Karl made the assumption that both were constant, and their answers are consistent with that assumption, however, all the other solutions could be correct by adjusting the snowfall rate and/or the plough rate. Now perhaps what I said may sound like quibbling, but I think it would be a lot more meaningful, and more fun, to try and solve problems that are well thought out, and consistent and have sufficient information.

My second point, is, that given the assumption that the snowfall and the plough rates are constant, the solutions given by Chris and Karl seem overly complicated. Here is another solution.

The quantity of snow removed by the plow in the first hour is q(t+1/2).

The quantity of snow removed by the plough in the next 3 hours is q(t + 1 + 3/2).

Since the plough rate is constant throughout then we can write

(t+1/2)/1 = (t + 1 + 3/2)/3 which simplifies to 2t=1, and t=1/2. Thus the snow started at 11:30.

By the way, I’m not sure whether its plow or plough, but I believe it should be spelled , and not spelt (unless of course you are from England or a time traveler from the 18th century).

December 20th, 2010 at 3:41 am

I like Al Gelman’s simplified approach of taking the average depth of snow in each time period and going from there. It definitely beats calculus! I was groping towards something like that myself but never came close.

I had no problem with snowfall rate and plough rate not being explicitly stated in the problem statement. It was reasonable to assume that both were constant since nothing was stated to the contrary.

December 20th, 2010 at 5:58 am

Hi AL. Making the assumptions is part of the fun of solving the problem.

I like your statement of assumption that the rate of ploughing is constant. In fact it leads to the speed of the plough being inversely proportional to the depth of the snow.

Please explain how you derived your equations.

December 20th, 2010 at 7:42 am

Hi Al. Yup – 18th century english. Chris worked out my snow plough could go ftl, and I am now stuck in the 18th Century, and English.

Further to Chris’s post 18 – many of the puzzles on this site do require a suspension of natures laws, reality etc and making a few assumptions along the way.

December 20th, 2010 at 1:54 pm

Karl, I really don’t see these puzzles as “a suspension of nature’s laws”. Far from it. I think they’re no different to what we have in the field of applied mathematics where phenomena are analysed in isolation to arrive at simplified formulae. These can then be modified as required to take account of other factors.

Chris, I gave a clue to the way Al derived his elegantly simple equations in my post # 17. What is the average depth of snow in each section of the plough’s journey?

I love these situations where simple approaches negate the need for calculus!

December 20th, 2010 at 2:19 pm

Well I seem to be a minority of one with my complaining about the presentation of the problems, so I will shut up about it.

Here is how I obtained those equations. When the plough arrived at the scene at 12:00 pm, it had been snowing for t hours at the constant rate of q liters/hour. So the amount of snow that had accumulated was q(t). Then the plowing began. It took an hour to clear 2 miles. The question is how much snow fell on the road during that 1 hour period? And the answer is q(1)/2 ( because the rate is assumed constant). So the total amount of snow to be removed is q(t + ½). When the plough gets to the next 2 miles, the amount of snow to be removed is q(t) + q(1). But of course it is still snowing and during the next 3 hours the amount of snow that will accumulate is q(3)/2 . so the total amount to be removed during the 3 hour period is q(t + 1 + 3/2). Now we assume that the rate of removal is constant, say Q liters/hour, and Q = quantity to be removed/time it takes to do so. In the first hour we must remove q(t+1/2) , the removal rate is therefore q(t+1/2)/1. in the next 3 houres we remove q(t+1+3/2), so the rate is q(t+1+3/2)/3. but since the rate of removal is constant, we can write q(t + ½)/1 = q(t+1+3/2)/3.

December 20th, 2010 at 3:45 pm

Of course – unfortunately – after the plough man has finished his 4 hours of work, there is still quite a bit of snow on the road. The first two miles will have accumulated an additional q(3+1/2), and the second two miles will have q(3/2).

How many hours do you think it will take him to do the next 2 miles?

On another topic entirely. How does one submit a problem for solution? I have a couple that I would like to share.

December 21st, 2010 at 2:12 am

Hi Wizard of Oz, I was just implying that snow rarely falls at a constant rate, the insects on the ruler rarely all travel at a uniform speed etc etc.

December 21st, 2010 at 4:44 pm

logically, the answer should be 10 am because it took untill 4 pm for the plow advanced 2 miles, so going back 3 hours, it should be 10 am.

December 22nd, 2010 at 6:53 am

Hi Al & Wiz. Sorry, I was being a bit of a twit. I don’t know why I couldn’t get my head round Al’s equations. I agree, they are correct, and that solution is very nice.

But a FTL plough, hardly a reasonable assumption (by us)

December 22nd, 2010 at 7:02 am

Hi WIz, while I’m at it, I couldn’t agree more; if the problem can be solved without calculus then that’s good.

That reminds me that Archimedes found the volume and surface area of the sphere (and the volume of the Steinmetz solid) without calculus; although he did use ideas that are a fore-runner of integral calculus, but are essentially intuitive ides.

December 22nd, 2010 at 1:48 pm

its both

December 23rd, 2010 at 9:01 am

Chris – Archimedes found the volume of the Steinmetz solid without calculus… did he tell Steinmetz?

I feel that this is proof of the ftl snow plough;-)

December 23rd, 2010 at 6:59 pm

What in the world is a Steinmetz solid?

December 27th, 2010 at 11:43 am

It would be sometime before noon because it never said the exact time and there is no way you woulod be able to figure that out.

December 27th, 2010 at 11:48 am

11 AM…

December 30th, 2010 at 6:03 pm

You know folks, it is great you solved this, because a mathematician I am not.

However, English I do enjoy, and I am a bit insulted by Plough and Plow comparisons, especially when my spell checker rejects “Plough” as a word. Where dig you folks study? LOL

Plow is the acceptable American English word for any piece of machinery that propels snow or other obstructions out of ones way or or prepares soil for planting.

I am partial to Plow as the word of choice, just so you all know, LOL

December 30th, 2010 at 6:18 pm

Oh my, now I see a typo in there, LOL I am so anal about such things and I make no secret about that, he he.

I meant, *did you folks study? Not DIG…LOL I am a barbarian for certain!

I don’t want to say I claim to be perfect, I am far from it, but I am mortified when I make a typo AND I catch it! LOL

If I don’t see it, it doesn’t count! LOL Isn’t that the way it should work?

I am only posting this since the puzzle is already solved, otherwise I may have kept it ALL to myself and I am sure we would have all been happier for it. LOL

Goodnight folks and sweet dreams. ;0} (A bear smile for you all)