2.
Given a logistic function with the following ordered pairs: (0, 10000), (10, 20000), & (20, 30000).
a) Find k.
10000 =
2=
L
10000 + 10000C
L
: L = 10000(1 + C ) = 10000 + 10000C : 20000 =
=
−
10
k
1+ C
1 + Ce
1 + Ce −10t
1+ C
1 + Ce
−10 k
(
10000 +
30000 =
2+
10000
3e −20 k − 1
1 − 2e −10 k
c)
1 − 2e −10 k
1
= 0 : Graph the function using x for k and find the x-intercept.
Find C.
C=
1+
1
1
3e −20 k
1 − 2e −10 k : 3 =
1 − 2e −10 k
: 3+
= 1+
1
1
⎛
⎞ −20 k
⎛
⎞ −20 k
1 − 2e −10 k
1 − 2e −10 k
1 + ⎜
e
1
+
e
⎟
⎜
⎟
⎝ 1 − 2e −10 k ⎠
⎝ 1 − 2e −10 k ⎠
k = 0.10986
b)
)
: 2 + 2Ce −10 k = 1 + C : 2Ce −10 k − C = −1 : C 2e −10k − 1 = −1 : C =
1
1 − 2e −10 k
= 3
Find L.
L = 10000(1 + C ) = 10000(1 + 3) = 40000
*