Snowplow problem
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5 members have voted
July 25th, 2020 at 3:21:49 PM
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One morning it starts to snow at a constant rate. Later, at 6:00am, a snowplow sets out to clear a straight street. The plow can remove a fixed volume of snow per unit time, in other words its speed it inversely proportional to the depth of the snow. If the plow covered 10 miles from 6 to 7am and 5 miles from 7 to 8am, what time did it start snowing?
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
July 25th, 2020 at 3:37:56 PM
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Let m be the number of minutes before 6 AM when it started snowing
The amount of snow cleared in the first 10 miles = 10m + 5 x 60
The amount of snow cleared in the next 5 miles = 5 (m + 60) + 5/2 x 60
10m + 300 = 5m + 300 + 150
m = 30, so it started snowing at 5:30 AM
July 25th, 2020 at 4:41:38 PM
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Quote: Wizard...what time did it start snowing?...
that it started snowing at approximately 5:23 AM.
I see that the answer has something to do with the golden ratio--the snow started before 6:00 AM by one hour divided by the golden ratio.
I see that the answer has something to do with the golden ratio--the snow started before 6:00 AM by one hour divided by the golden ratio.
July 25th, 2020 at 4:46:19 PM
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Forgot way to much to solve it the intended way tonight...
But my excel-solution seems to converge towards (phi-1) hours before 6
So i guess it started 06:22:55
6:30 is for sure not correct
But my excel-solution seems to converge towards (phi-1) hours before 6
So i guess it started 06:22:55
6:30 is for sure not correct
July 25th, 2020 at 5:19:40 PM
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Quote: ChesterDogthat it started snowing at approximately 5:23 AM.
I see that the answer has something to do with the golden ratio--the snow started before 6:00 AM by one hour divided by the golden ratio.
I agree. Remember, for full credit you must show your work.
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
July 25th, 2020 at 5:57:16 PM
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A try..
I assume it started to snow at time 0:
The plow have a speed at 1/X
so f(X) = 1/x
We have that the definite integral from x to (x+1) = 2 times the definite integral from (x+1) to (x+2)
The integral for 1/x = ln(x)
That gives:
ln(x+1) - ln(x) = 2*( ln(x+2) - ln(x+1) )
3*ln(x+1) -ln(x) = 2*ln(x+2)
3*ln(x+1) = 2*ln(x+2)+ln(x)
ln((x+1)^3) = ln(x) + ln((x+2)^2)
ln((x+1)^3) = ln((x+2)^2 * x)
ln(x^3+3x^2+3x+1)= ln(x^3+4x^2+4x)
x^3+3x^2+3x+1= x^3+4x^2+4x
x^2 +x-1 = 0
X = -1,618 and 0,618
As it started to snow before the plow started, the solution have to be positive and the answer is 0,618.
=it started to snow at 06:22:55
Sorry my poor english and formating.
July 25th, 2020 at 7:34:30 PM
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Quote: TorghattenForgot way to much to solve it the intended way tonight...
But my excel-solution seems to converge towards (phi-1) hours before 6
So i guess it started 06:22:55
6:30 is for sure not correct
Quote: Torghatten
A try..
I assume it started to snow at time 0:
The plow have a speed at 1/X
so f(X) = 1/x
We have that the definite integral from x to (x+1) = 2 times the definite integral from (x+1) to (x+2)
The integral for 1/x = ln(x)
That gives:
ln(x+1) - ln(x) = 2*( ln(x+2) - ln(x+1) )
3*ln(x+1) -ln(x) = 2*ln(x+2)
3*ln(x+1) = 2*ln(x+2)+ln(x)
ln((x+1)^3) = ln(x) + ln((x+2)^2)
ln((x+1)^3) = ln((x+2)^2 * x)
ln(x^3+3x^2+3x+1)= ln(x^3+4x^2+4x)
x^3+3x^2+3x+1= x^3+4x^2+4x
x^2 +x-1 = 0
X = -1,618 and 0,618
As it started to snow before the plow started, the solution have to be positive and the answer is 0,618.
=it started to snow at 06:22:55
Sorry my poor english and formating.
Really? Twice you have him plowing BEFORE it starts snowing??? I think you missed a digit or something.
This math is was beyond my abilities, but I know that’s wrong.
I invented a few casino games. Info:
http://www.DaveMillerGaming.com/ —————————————————————————————————————
Superstitions are silly, irrational, childish rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
July 25th, 2020 at 8:19:16 PM
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Quote: Torghatten
A try..
I assume it started to snow at time 0:
The plow have a speed at 1/X
so f(X) = 1/x
We have that the definite integral from x to (x+1) = 2 times the definite integral from (x+1) to (x+2)
The integral for 1/x = ln(x)
That gives:
ln(x+1) - ln(x) = 2*( ln(x+2) - ln(x+1) )
3*ln(x+1) -ln(x) = 2*ln(x+2)
3*ln(x+1) = 2*ln(x+2)+ln(x)
ln((x+1)^3) = ln(x) + ln((x+2)^2)
ln((x+1)^3) = ln((x+2)^2 * x)
ln(x^3+3x^2+3x+1)= ln(x^3+4x^2+4x)
x^3+3x^2+3x+1= x^3+4x^2+4x
x^2 +x-1 = 0
X = -1,618 and 0,618
As it started to snow before the plow started, the solution have to be positive and the answer is 0,618.
=it started to snow at 06:22:55
Sorry my poor english and formating.
Great answer and solution Tor! I don't recall you being a member of the Beer Club yet, in which case welcome. This means I owe you a beer upon your next visit to Vegas.
To be perfectly honest, I asked this problem because I made a video going over the solution. This is my first math video in this kind of format so please go easy on me with your reviews.
Direct: https://www.youtube.com/watch?v=33KYd7OSKjI
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
July 25th, 2020 at 10:02:24 PM
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July 26th, 2020 at 6:12:47 AM
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Quote: DJTeddyBear
Really? Twice you have him plowing BEFORE it starts snowing??? I think you missed a digit or something.
This math is was beyond my abilities, but I know that’s wrong.
Sorry, misread and used 07-09 instead of 06-08.
Quote: Wizard
Great answer and solution Tor! I don't recall you being a member of the Beer Club yet, in which case welcome. This means I owe you a beer upon your next visit to Vegas.
To be perfectly honest, I asked this problem because I made a video going over the solution. This is my first math video in this kind of format so please go easy on me with your reviews.
Direct: https://www.youtube.com/watch?v=33KYd7OSKjI
Thx :)
Idk when im able/allowed to go to Vegas again, but most likely it will be next summer :p
