This preview shows pages 1–3. Sign up to view the full content.
Garcia, Ilse – Homework 14 – Due: Nov 27 2007, 3:00 am – Inst: Fonken
1
This printout should have 10 questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
be±ore answering.
The due time is Central
time.
001
(part 1 o± 1) 10 points
Scientists began studying the elk popula
tion in Yellowstone Park in 1990 when there
were 700 elk. They determined that
t
years
a±ter the study began the population size,
N
(
t
), was increasing at a rate proportional to
1200

N
(
t
).
I± the population was 1100 in
year 2000, estimate the size o± the elk popula
tion in year 2010?
1.
population size
≈
1120
2.
population size
≈
1140
3.
population size
≈
1180
correct
4.
population size
≈
1160
5.
population size
≈
1200
Explanation:
The population size,
N
(
t
), satisfes the
equations
dN
dt
=
k
(1200

N
)
,
N
(0) = 700
with
k
a constant. Thus

ln(1200

N
) =
kt
+
C
which a±ter exponentiation becomes
N
(
t
) = 1200

Ae

kt
where
A
is an arbitrary constant. Now
A
is
determined by the initial condition on
N
(
t
)
because
N
(0) = 700
=
⇒
A
= 500
.
Consequently,
N
(
t
) = 1200

500
e

kt
.
On the other hand, the value o±
k
is deter
mined by the population size in year 2000
when
N
(10) = 1100, ±or then
100 = 500
e

10
k
.
Taking logs o± both sides we thus see that
k
=
1
10
ln
‡
5
·
.
In year 2010, there±ore,
N
(20) = 1200

500
e

2 ln 5
.
Hence, in year 2010,
population size
≈
1180
.
keywords:
002
(part 1 o± 1) 10 points
An investor buys shares o± a stock ±or $2000
and sells them 10 years later ±or $4000. I± in
terest is compounded continuously, determine
the annual % rate o± interest that the original
$2000 investment earned, leaving your answer
in logarithmic ±orm.
1.
interest rate =
µ
10 ln
3
2
¶
%
2.
interest rate =
µ
10 ln
7
4
¶
%
3.
interest rate =
µ
1
10
ln
9
4
¶
%
4.
interest rate =
µ
10 ln
9
4
¶
%
5.
interest rate =
µ
1
10
ln 2
¶
%
6.
interest rate =
µ
1
10
ln
3
2
¶
%
7.
interest rate = (10 ln 2) %
correct
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentGarcia, Ilse – Homework 14 – Due: Nov 27 2007, 3:00 am – Inst: Fonken
2
8.
interest rate =
µ
1
10
ln
7
4
¶
%
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 schultz
 Differential Calculus

Click to edit the document details