If the relationship is proportional, what is the missing value from the table
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Let denote the amount of salt in the tank at time
. We're told that
.
For , the salt flows in at a rate of (1/5 lb/gal)*(5 gal/min) = 1 lb/min. When the regulating mechanism fails, 20 lbs of salt is dumped and no more salt flows for
. We can capture this in terms of the unit step function
and Dirac delta function
as
(in lb/min)
The salt from the mixed solution flows out at a rate of
Then the amount of salt in the tank at time changes according to
Let denote the Laplace transform of
,
. Take the transform of both sides to get
Solve for , then take the inverse of both sides.