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)
Pretend the snow falls discretely over x even intervals of distance plowed.
(For example if x=10 after the first 5 miles takes .1 hours, .1 inch falls. Since there are 1.1 inches on the ground the next 5 miles takes an additional .11 hour so .11 inch falls etc.)
The time, in terms of x, is ((x+1)/x)^x - 1 or
t(x)=(1+1/x)^x - 1.
What is sought is the limit as x goes to infinity. Since e is defined as the limit as x goes to infinity of (1+1/x)^x the solution is e-1 or about 1.718 hours.
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