document.write( "Question 1206194: I'm not sure how I would solve this; I have some idea that there is a d=r*t equation in this but not sure where do start. Here's what I have:
\n" ); document.write( "Early one morning it starts to snow at a constant rate.\r
\n" ); document.write( "\n" ); document.write( "Later, at 6:00 AM, a snow plow sets out to clear a straight street.\r
\n" ); document.write( "\n" ); document.write( "The plow can remove a fixed volume of snow per unit of time.\r
\n" ); document.write( "\n" ); document.write( "In other words, the speed of the plow is inversely proportional to the depth of the snow.\r
\n" ); document.write( "\n" ); document.write( "If the plow covered twice as much distance in the first hour as the second hour, what time did it start snowing? \n" ); document.write( "

Algebra.Com's Answer #843465 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Early one morning it starts to snow at a constant rate.
\n" ); document.write( "Later, at 6:00 AM, a snow plow sets out to clear a straight street.
\n" ); document.write( "The plow can remove a fixed volume of snow per unit of time.
\n" ); document.write( "In other words, the speed of the plow is inversely proportional to the depth of the snow.
\n" ); document.write( "If the plow covered twice as much distance in the first hour as the second hour, what time did it start snowing?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        There are two approaches (two different levels of complexity of a model consideration).
\n" ); document.write( "        One approach is Algebra, assuming using discrete time and discrete height of the snow in time.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Another approach is differential equation, assuming continuous time and continuous function of the height of the snow in time.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        First approach is much easier (it gives an approximate solution, but uses simple idea and simple solution technique).\r
\n" ); document.write( "\n" ); document.write( "        Second approach assumes using and solving ordinary differential equations.
\n" ); document.write( "        Second approach gives an exact solution, which may be different from the Algebra solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Here I present an approximate Algebra solution, simple and elegant.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let m be the number of minutes before 6 AM when it started snowing;\r\n" );
document.write( "    r be the rate of snowing;\r\n" );
document.write( "    2L be the distance cleared in the first hour;\r\n" );
document.write( "     L be the distance cleared in the second hour.\r\n" );
document.write( "\r\n" );
document.write( "The amount of snow cleared in the first hour = 2L*r(m+30).\r\n" );
document.write( "The amount of snow cleared in the second hour = Lr*((m + 30) + 60).\r\n" );
document.write( "\r\n" );
document.write( "The rate of removing snow is the same; so we can write this equation\r\n" );
document.write( "\r\n" );
document.write( "    2L*r(m+30) = Lr*(m+30+60).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Here left side is the volume (the area of vertical cross-section) of the snow \r\n" );
document.write( "on the part of the road of the length 2L at the time moment (m+30) minutes. \r\n" );
document.write( "This time moment is the mid of the first hour.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Right side is the volume (the area of vertical cross-section) of the snow \r\n" );
document.write( "on the part of the road of the length L at the time moment (m+30+60) minutes, \r\n" );
document.write( "i.e. one hour later.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Cancel Lr in both sides and get\r\n" );
document.write( "\r\n" );
document.write( "    2m + 60 = m + 90\r\n" );
document.write( "\r\n" );
document.write( "From it,  m = 90 - 60 = 30.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this way, we get the ANSWER :  it started snowing at 5:30 am.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( ". . . . . . . . . . . .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In the solution above, I used a rectangular approximation for the volume/(vertical section) \r\n" );
document.write( "of the snow in the road. \r\n" );
document.write( "\r\n" );
document.write( "Let's try a trapezoid approximation for the volume /(vertical section) of the snow in the road.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the equation is\r\n" );
document.write( "\r\n" );
document.write( "     = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It gives, after canceling Lr in both sides\r\n" );
document.write( "\r\n" );
document.write( "    m + (m+60) = m + 90\r\n" );
document.write( "\r\n" );
document.write( "    2m + 60 = m + 90\r\n" );
document.write( "\r\n" );
document.write( "     m = 90 - 60 = 30.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "With this approximation, we get the same answer.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "////////////////\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This problem is FAMOUS.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "See the links to other sites in the Internet to similar (not necessary identical) problems.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://mindyourdecisions.com/blog/2018/04/05/this-is-not-a-trick-question-the-famous-snow-plow-math-problem/\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://wizardofvegas.com/forum/questions-and-answers/math/34930-snowplow-problem/ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://personal.math.ubc.ca/~israel/m215/plows/plows.html\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://sites.science.oregonstate.edu/~show/docs/256-SnowPlow.pdf\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );