Hi. Kindly provide me solutions for these. Thanks! May the force be with you!AMAT 110
Exam 2
Mathematical Modelling
Due: 12 May, 5PM
General Directions: Do not consult anyone except your teacher. Show clear, complete
and concise solutions to earn full points. Show strictly handwritten answers. Submit
your answers on sheets of paper, double-spaced.
1. A patient is given a dosage
Q
of a drug at regular intervals of time
T
. The concentration of
the drug in the blood has been shown experimentally to obey the law
dC
dt
=
-
ke
C
.
(a) If the ?rst does is administered at
t
= 0 hours, show that after
T
hours have elapsed, the
residual
R
1
=
-
ln
(
kT
+
e
-
Q
)
remains in the blood.
(b) Assume an instantaneous rise in concentration whenever the drug is administered. Show
that after the second dose and
T
hours have elapsed again, the residual
R
2
=
-
ln
(
kT
(
1 +
e
-
Q
)
+
e
-
2
Q
)
remains in the blood.
(c) Show that the limiting value
R
of the residual concentrations for doses of
Q
mg/ml
repeated at intervals of
T
hours is given by
R
=
-
ln
kT
1
-
e
-
Q
.
(d) Assuming an ine?ective lower level of concentration
L
and harmful upper level at some
higher concentration
H
, show that the dose schedule
T
for a safe and e?ective concen-
tration of the drug in the blood satis?es
T
=
1
k
(
e
-
L
-
e
-
H
)
where
k
is a positive constant.
2. It is know that if the tamaraw population
P
falls below a certain level
m
, the tamaraw will
become extinct. In addition, if the tamaraw population rises above the carrying capacity
M
,
the population will decrease back to
M
through disease and malnutrition
(a) Discuss the reasonableness of the following model for the growth rate of the tamaraw
population as a function of time:
dP
dt
=
rP
(
M
-
P
) (
P
-
m
)
where
P
is the tamaraw population and
r
is a positive constant of proportionality. Include
a phase line.
(b) Show that if
P > M
, then lim
t
??
P
(
t
) =
M
. (
Hint
: use the population curve.)
(c) What happens if
P
This question was answered on: Sep 21, 2023
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