(Solution) AMAT 110 Exam 2 Due: 12 May, 5PM Mathematical Modelling General Directions: Do not consult anyone except your teacher. Show clear, complete and... > Snapessays.com

(Solution) AMAT 110 Exam 2 Due: 12 May, 5PM Mathematical Modelling General Directions: Do not consult anyone except your teacher. Show clear, complete and...

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

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

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