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Slowing by snowing (Posted on 2010-09-13) Difficulty: 2 of 5
A snowplow heads out to plow a 50 mile stretch of highway when there is already an inch of snow on the ground. The snow is falling steadily at a rate of one inch per hour. The speed of the plow is 50/d mph where d is the depth of the snow in inches.

How long does the plow travel this 50 mile stretch of road? (one way)

See The Solution Submitted by Jer    
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solution by integration | Comment 3 of 6 |
In order to find the time traveled by the plow, we must find some value, T, such that 50 = ∫(S) dt evaluated from 0 to T, where S is our speed function.

d = d0 + r * t --> d = 1 + t where d0 = 1 inch, and r = 1 inch / hr

S = 50 / d --> S = 50 / (1 + t)

50 = ∫(50 / (1 + t)) dt = 50 ∫1 / (1 + t) dt

Dividing both sides by 50, gets us:

1 = ∫[1 / (1 + t)] dt ... since our function to be integrated is of the form (1 / u) du, we know the integral is ln(u).

So, we now have 1 = ln(1 + t) evaluated from 0 to T, where T is the total time traveled.

1 = ln(1 + T) - ln(1 + 0) --> 1 = ln(1 + T) - ln(1) -->

1 = ln(1 + T) - 0

Taking e to the power of both sides, we can eliminate the natural log:

e = 1 + T ... T = e - 1 -->

T = 1.718281828459045 hours or T = 6185.814582452563 seconds

  Posted by Justin on 2010-09-13 15:24:37
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