# CAS MA 124 Study Guide - Midterm Guide: Ion, Horse Length, Partial Fraction Decomposition

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10 Oct 2018

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MA 124 - Fall ’12 - Exam 1A - 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, iodine has a half-life in the bloodstream

of 8 days.

a) A patient is given an initial dose of 80 millicuries. Find the function that gives the

amount in the body after tdays.

b) How long does it take for the amount of iodine to reach 20% of the original dose ?

ANSWER: a)

A(t)=A0e−kt = 80e−kt

Need k

Using the formula

k=ln 2

t1/2

=ln 2

8

which can be obtained by

A(8) = 40 = 80e−k8⇒1

2=e−k8

ln 1

2=lne−k8=−k8⇒k=−ln 1

2

8=ln 2

8

So

A(t) = 80e−

ln 2

8t

b) Solve

.2(80) = 16 = 80e−

ln 2

8t

I.e.

.2=e−

ln 2

8t

ln(.2) = ln e−

ln 2

8t=−ln 2

8t

So

t=−8ln(.2)

ln 2 =−8 ln(1/5)

ln 2 =8ln5

ln 2

2) a) (8 points) Find

�e

1

xln xdx

ANSWER a) Use integration by parts with

u=lnxdv=xdx

du =1

xdx v =�xdx=1

2x2

�udv=uv −�vdu

gives

�xln xdx=1

2x2ln x−1

2�xdx

=1

2x2ln x−1

2·1

2x2

=1

2x2ln x−1

4x2

So �e

1

xln xdx=1

2e2ln e−1

4e2−(1

2ln 1 −1

4)

=1

2e2−1

4e2−(−1

4)

=1

4e2+1

4

=1

4(e2+ 1)