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Chapter 3
A
=
1
(6000)7
ln
±
5000(3000
−
1000)
1000(5000
−
3000)
²
=
ln 5
42000
.
Thus the formula (15) on page 95 of the text becomes
p
(
t
)=
p
0
p
1
p
0
+(
p
1
−
p
0
)
e
−
Ap
1
t
=
(1000)(6000)
(1000) + (6000
−
1000)
e
−
(ln 5
/
42000)6000
t
=
6000
1+5
1
−
t/
7
.
(3.1)
In the year 2010,
t
= 2010
−
1980 = 30, and the estimated population of splake is
p
(30) =
6000
1
−
30
/
7
≈
5970
.
Taking the limit in (3.1), as
t
→∞
, yields
lim
t
→∞
p
(
t
) = lim
t
→∞
6000
1
−
t/
7
=
6000
1 + lim
t
→∞
5
1
−
t/
7
= 6000
.
Therefore, the predicted limiting population is 6000.
15.
Counting time from the year 1970, we have the following data:
t
0
=0
,p
0
=
p
(
t
0
) = 300;
t
a
= 1975
−
1970 = 5
a
=
p
(
t
a
) = 1200;
t
b
= 1980
−
1970 = 10
b
=
p
(
t
b
) = 1500
.
Since
t
b
=2
t
a
, we use the formulas in Problem 12 to Fnd parameters in the logistic model.
p
1
=
±
(1200)(1500)
−
2(300)(1500) + (300)(1200)
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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