Question

Morphine is administered to a patient intravenously at a rate of 2.5 mg per hour. About 34.7% of the morphine is metabolized and leaves the body each hour.

M'(t)=34.7% - (2.5*M(t))

If there is initially 100mg of morphine in the body, how much morphine will there be after 5 hours?

Answer 1

differential calculus ?

Answer 2

yeah i'm in calc 1

Answer 3

if we use y (instead of M)
\[ \frac{dy}{dt} = .347 - 2.5 y \\
\frac{\frac{dy}{dt} }{ .347 - 2.5 y }=1 \\
\frac{dy}{.347 - 2.5 y }=dt\]

Answer 4

we can integrate both sides
\[ -\frac{2}{5} \int \frac{ - 2.5 \ dy}{0.347 - 2.5y}=\int dt \]

Answer 5

can you integrate that ?

Answer 6

where did 2/5 come from?

Answer 7

if we make the bottom u
u = 0.347 - 2.5y
then
du = -2.5 dy

and we would have du/u
...which integrates to ln( u )

so I put in a -2.5 i.e. -5/2 up top
and a -2/5 out front to compensate

Answer 8

oh ok

Answer 9

i'm forgetting how to integrate I haven't done it in a while XD

Answer 10

or you could do a "change of variables"

(which is really the same thing)

let u = 0.347 - 2.5 y
du = -2.5 dy= - 5/2 dy
dy = -2/5 du

and replace dy/(0.347 -2.5y) with -2/5 du / u

Answer 11

Yeah I get that

Answer 12

but now i'm confused as to what I do next? I know I leave the -2/5 alone for now

Answer 13

\[ \int \frac{dx}{x} = \ln x +C\]

Answer 14

so is the x in lnx the -2.5dm / (.347-2.5m) ?

Answer 15

the "x" would be the entire bottom
dx is -2.5 dm

Answer 16

so -2.5dm/(.347-2.5m) = ln(.347-2.5m) + C ??

Answer 17

and what happens with the -2/5 ?

Answer 18

before going on, are you sure about that equation, because it's not making sense to me
I'm thinking it would be
M' = -0.347M + 2.5

Answer 19

maybe it is. i had to come up with the equation but i wan't sure about it!

Answer 20

yours makes more sense because i though we would have to subtract a number from the equation, but it makes more sense to add the 34.7%

Answer 21

M' = 2.5
says: the change in morphine is +2.5 mg (per hour)
assuming it was not metabolized. i.e. it goes up at that rate
but some of it "disappears"
how much ? 34.7% of how much there is i.e. 34.7% of M

Answer 22

so the total change is
M' = 2.5 - 0.347M

that says we add 2.5 mg, but subtract off 34.7% of what's in the body

Answer 23

oh i understand!

Answer 24

could i use that equation to find the amount of morphine after 5 hours?

Answer 25

now we solve it the same as before.
write it as
\[ \frac{ \frac{dM}{dt}}{2.5- 0.347M} = 1\]
multiply both sides by "dt" (this is a dubious step mathematically , but we are "separating the variables")
we get
\[ \frac{dM}{2.5 - 0.347 M } = dt \]

Answer 26

multiply both sides by -0.347
(this will make the top the derivative of the bottom so that we have a dx/x form)

Answer 27

dt(-.347)=dm/(2.5-m) ??

Answer 28

your algebra is weak ? (calculus assumes you know algebra very well)

\[ -0.347 \ dt = \frac{-0.347 \ dm}{2.5- 0.347 m} \]

Answer 29

now integrate both sides
\[ -0.347 \int \ dt = \int \frac{-0.347 \ dm}{2.5- 0.347 m} \\
-0.347 \ t+C = \ln \left(2.5- 0.347 m \right)

\]

Answer 30

and then i bring everything to one side right?

Answer 31

we could put the unknown constant of integration on either side.
make each side the exponent of "e" (to get rid of the natural logarithm)
we get
\[ e^{-0.347 \ t + C} = 2.5 - 0.347 m \]
or, writing e^C as an (unknown) constant A
\[ A e^{-0.347 \ t}= 2.5 - 0.347 m \]
now solve for m

Answer 32

i can use either equation?

Answer 33

\[ A e^{-0.347 t} - 2.5 = - 0.347 m \\
m = A e^{-0.347 t} + \frac{2.5}{0.347} \]

Answer 34

notice when we divide A by -0.347 we just keep the answer A
(A is arbitrary, and after dividing it's still arbitrary)

Answer 35

but if i used the 1st equation would i divide the whole thing by -.347?

Answer 36

we get roughly
\[ m = A e^{-0.347 t} + 7.205\]
at t=0 we have 100 mg
putting in t=0 and m=100 we get
100= A+7.205
A= 100-7.205 = 92.8 (roughly)
and finally
\[ m= 92.8e^{-2.5 t} +7.205 \]

Answer 37

** the last equation should read
\[ m = 92.8 e^{-0.347 t} + 7.205 \]

Answer 38

oh ok that's what i was confused about XD

Answer 39

m=23.6 (rounded)

Answer 40

yes, they don't say how accurate they want the answer, but that looks good.

Answer 41

ok thank you so much! this makes much more sense