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)
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
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Posted by Justin
on 2010-09-13 15:24:37 |
solution by integration
