SOLUTION: Find the logistic function that satisfies the given conditions: Initial value=5, limit to growth=20, passing through (1,8)

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Question 1206367: Find the logistic function that satisfies the given conditions: Initial value=5, limit to growth=20, passing through (1,8)
Answer by MathLover1(20850) About Me  (Show Source):
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Find the logistic function that satisfies the given conditions.

given:
Initial value = 5
limit to growth = 20,
passing through (1, 8)

Use the logistic model:

f%28x%29=c%2F%281%2Ba%2Ab%5Ex%29
f%280%29=5=> at (0,5)

limit to growth => c=20

5=20%2F%281%2Ba%2Ab%5E0%29
5=20%2F%281%2Ba%2A1%29
5=20%2F%281%2Ba%29
5%281%2Ba%29=20
5%2B5a=20
5a=20-5
5a=15
a=3


so far:
f%28x%29=20%2F%281%2B3%2Ab%5Ex%29

if passing through (1, 8) , we have

8=20%2F%281%2B3%2Ab%5E1%29
8=20%2F%281%2B3%2Ab%29
8%281%2B3%2Ab%29=20
8%2B24b=20
24b=20-8
24b=12
b=12%2F24
b=1%2F2

=> f%28x%29=20%2F%281%2B3%2A%281%2F2%29%5Ex%29