I got a version of the snow plow problem for differential equations but I can't seem to solve it for the life of me. I looked at some solutions online and had trouble comprehending them. If some one could give me a hand it would be much appreciated.
The problem: One day snow began to fall before dawn and continued to fall at a constant rate. At 6AM a road crew set out to clear the roads. By 7AM they had cleared 5 miles. By 8AM they had cleared and additional 3 miles. When did the snow begin to fall.
What I've written so far:
I can't for the life of me figure out how to solve the system of equations that I've gotten. Thanks in advance for the help.
Differential Equations snow plow problem
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 Schmendreck
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 Joined: Wed Aug 22, 2007 6:18 pm UTC
 Location: New York
Differential Equations snow plow problem
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Robert A. Heinlein
Robert A. Heinlein
Re: Differential Equations snow plow problem
i cant read the doc file, but i would say about 3:33 AM
t time
a snowfall begin
s snow
c snowing rate
[math]s(t) = c \cdot (ta)[/math]
plowing ratio: 3/5
volume ratio: [math]\frac{ \int_a^7 s(t) dt}{\int_a^8 s(t) dt} = \frac{7² c/2  7ca ca²/2+ca²}{ 8²c/2 8ca ca²/2+ca²}[/math]
setting both ratios equal and dividing by c !=0 we get
1/5 a² + 11/5 a 53/10 =0
a_1 = 3,5635... < should be the one we want
a_2 = 7,436....
i hope i made no mistake but this looks quite right
t time
a snowfall begin
s snow
c snowing rate
[math]s(t) = c \cdot (ta)[/math]
plowing ratio: 3/5
volume ratio: [math]\frac{ \int_a^7 s(t) dt}{\int_a^8 s(t) dt} = \frac{7² c/2  7ca ca²/2+ca²}{ 8²c/2 8ca ca²/2+ca²}[/math]
setting both ratios equal and dividing by c !=0 we get
1/5 a² + 11/5 a 53/10 =0
a_1 = 3,5635... < should be the one we want
a_2 = 7,436....
i hope i made no mistake but this looks quite right
Re: Differential Equations snow plow problem
Ah yes this problem. I remember having to do this one about four months ago for my diff eq class. Anyway here are the solutions (part B is included).
Hope this helps.
BTW I got T=6.29 hours. Also it is important to note that the time must be negative since it began to snow before the snowplow left.
Hope this helps.
BTW I got T=6.29 hours. Also it is important to note that the time must be negative since it began to snow before the snowplow left.
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 MathProjectSnowplow.doc
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 Schmendreck
 Posts: 197
 Joined: Wed Aug 22, 2007 6:18 pm UTC
 Location: New York
Re: Differential Equations snow plow problem
I found an explanation of the correct solution.
http://www.as.ysu.edu/~faires/Courses/2 ... owplow.pdf
With this i get that the snow plow leaves about an hour and 5 minutes after it starts snowing.
http://www.as.ysu.edu/~faires/Courses/2 ... owplow.pdf
With this i get that the snow plow leaves about an hour and 5 minutes after it starts snowing.
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Robert A. Heinlein
Robert A. Heinlein
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Re: Differential Equations snow plow problem
Schmendreck wrote:With this i get that the snow plow leaves about an hour and 5 minutes after it starts snowing.
I get the same result (with the help of WolframAlpha to get the root of the horrible degree7 polynomial you end up with... I'd gotten the horrible polynomial, but figured there had to be a better way, until I read your PDF, saw that was the right way, and plugged it into Alpha).
It really should be stated in the question though, how the speed of the plow changes... I mean, assuming that the rate of snow moved by the plow is a constant (ie the speed of the plow is inversely proportional to the height of the snow) is all well and good, but it's just an assumption... it should be in the question. It seems a bit silly to assume that if it had left at 5AM instead of 6AM (when it had only been snowing for 5 minutes) it'd be initially moving at just over 88 miles an hour (maybe they did, and it travelled forward in time to 6AM?)
Is this really a differential equation, though? I mean, we find x'(t) in terms of t, and find x(t)... that's just integration. I thought differential equations were finding x'(t) in terms of x(t) and possibly t (or similar for higher orders)... Just a difference of terminology?
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