CAS MA 124 Study Guide - Midterm Guide: Diazepam, Partial Fraction Decomposition, Ion

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Published on 10 Oct 2018
Professor
MA 124 - Fall ’12 - Exam 1B - Answers
GENERAL INSTRUCTIONS: Show all your work. A correct answer without justifi-
cation receives no credit. Check your algebra and arithmetic carefully.
1) (16 points) When given as a medication, valium has a half-life in the bloodstream
of 10 days.
a) A patient is given an initial dose of 60 millicuries. Find the function that gives the
amount in the body after tdays.
b) How long does it take for the amount of valium to reach 30% of the original dose ?
ANSWER: a)
A(t)=A0ekt = 60ekt
Need k
Using the formula
k=ln 2
t1/2
=ln 2
10
which can be obtained by
A(10) = 30 = 60ek10 1
2=ek10
ln 1
2=lnek10 =k10 k=ln 1
2
10 =ln 2
10
So
A(t) = 60e
ln 2
10 t
b) Solve
.3(60) = 18 = 60e
ln 2
10 t
I.e.
.3=e
ln 2
10 t
ln(.3) = ln e
ln 2
10 t=ln 2
10 t
So
t=10 ln(.3)
ln 2
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2) (16 points) Find the volume of the solid obtained by rotating the region bounded
by
y=1
9x2,y=0,x=2,x=2
about the x-axis.
ANSWER: A sketch of the curve shows that the region sits on the x-axis and the
method of disks can be applied.
The disks have Area = πr2with
r=1
9x2
and
Area = π(1
9x2)2=π1
9x2
Since 2x2 we get
V=π2
2
1
9x2dx
NOTE: Since (9 (x)2)=9x2we can write this as
2π2
0
1
9x2dx
To do the above integral use partial fractions.
Since 9 x2=(3x)(3 + x)wewrite
1
9x2=A
3x+B
3+x
Multiplying both sides by 9 x2gives
1=A(3 + x)+B(3 x)
Substituting x= 3 gives 1 = 6AA=1/6.
Substituting x=3 gives 1 = 6BB=1/6.
So 1
9x2=1
6·1
3x+1
6·1
3+x
hence
2
0
1
9x2=1
62
0
1
3xdx +1
62
0
1
3+xdx
=1
62
0
1
3xdx +2
0
1
3+xdx
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For the first integral, letting u=3xgives du =dx. Also
x=0u= 3 and x=2u=1
Hence
2
0
1
3xdx =1
3
1
udu =3
1
1
udu
Integrating this gives
ln |u|
3
1=ln3ln 1 = ln 3
For the second integral, letting u=3+xgives du =dx. Also
x=0u= 3 and x=2u=5
Hence
2
0
1
3+xdx =5
3
1
udu
Integrating this gives
ln |u|
5
3=ln5ln 3
Combining our constants from above we get an answer of
2π(1
6)(ln 3 + (ln 5 ln 3)) = π
3ln 5
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