1. Dust is generated continuously in a room by an industrial process at the rate of .02 lbs/
min.
The volume of the room is 4000 cubic feet and you may assume that the air in the
room is well mixed by fans.
The air conditioning system draws air from the room at the
rate of r = 800 cubic feet per minute, and the filtered air is returned to the room at the
same rate.
The filter reduces the concentration of dust in the air passing through it by a
multiplicative factor of the form e
.2b
where b is the thickness in inches of the filter
material.
(a) Derive a differential equation which models the dynamical process described above.
(Hint: Choose the state variable to be d(t), the amount of dust in the room (in pounds) at
time t).
(b) Solve the differential equation derived in part (a) if the initial of dust in the room is
10
5
pounds per cubic foot.
(c) What is the minimal acceptable thickness of the filter if the limiting concentration of
dust in the room must not exceed 4.0
×
10
5
pounds per cubic foot.
2. A wellmixed tank initially contains 100 gal.of pure water in which 10 lbs. of salt are
dissolved.
Pure water runs into the tank at a rate of 1 gal/min.
The solution runs out at a
rate of 4 gal/min and is piped immediately to an evaporator which removes half of the
water without removing any of the salt.
Some of this more concentrated solution is
returned immediately to the tank at a rate of 1 gal/min.
The rest of this more
concentrated water is discarded as waste.
(a)
Assuming that this recycling is instantaneous, derive a differential equation and
initial condition for the amount of salt (in lbs.) S(t) in the tank at time t (in min.).
(b)
For what values of t is your equation valid?
(c)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Dapkus
 Exponential Function

Click to edit the document details