3.7. Logistic Functions http://www.ck12.org
2 , 000 , 000 = 120 +, 1000. 5 ·,b^000 − 100
1 + 1. 5 ·b−^100 = 10
b−^100 = 19. 5 = 6
b≈ 0. 98224The question asks for thexvalue whenf(x) = 10 , 000 ,000.
10 , 000 , 000 = 1 + 120. 5 ,^000 ·( 0 ., 98224000 )x
1 + 1. 5 ·( 0. 98224 )x= 2
( 0. 98224 )x=^23
x·ln( 0. 98224 ) =ln( 2
3
)
x= ln( 2
3)
ln( 0. 98224 )≈^22.^629This means that according to your assumption and the two population data points you used, the predicted time from
now that the population of Long Island will reach 10 million inhabitants is about 22.6 years.
Example C
A special kind of algae is grown in giant clear plastic tanks and can be harvested to make biofuel. The algae are given
plenty of food, water and sunlight to grow rapidly and the only limiting resource is space in the tank. The algae are
harvested when 95% of the tank is full leaving the tank 5% full of algae to reproduce and refill the tank. Currently
the time between harvests is twenty days and the payoff is 90% harvest. Would you recommend a more optimal
harvest schedule?
Solution: Identify known quantities and model the growth of the algae.
Known quantities:( 0 , 0. 05 );( 20 , 0. 95 );c= 1 or100%
0. 05 = 1 +^1 a·b 0
a= 19
0. 95 = 1 + 191 ·b 20
1 + 19 ·b^20 = 0.^195b^20 =( 1
0. 95 −^1
)
19
b≈ 0. 74495The model for the algae growth is:
f(x) = 1 + 19 ·( 01. 74495 )x
The question asks about optimal harvest schedule. Currently the harvest is 90% per 20 day or a unit rate of 4.5% per
day. If you shorten the time between harvests where the algae are growing the most efficiently, then potentially this