Answer
(a) On the graph, we can see that the population reaches 900 after approximately 4.5 years.
(b) This function tells us how much time is required (in years) for a population to reach a level of $P$.
(c) It takes 4.4 years for the population to reach 900.
Work Step by Step
(a) On the graph, we can see that the population reaches 900 after approximately 4.5 years.
(b) We can find the inverse:
$P = \frac{100,000}{100+900e^{-t}}$
$100+900e^{-t} = \frac{100,000}{P}$
$900e^{-t} = \frac{100,000}{P}-100$
$e^{-t} = \frac{1}{900}\cdot(\frac{100,000}{P}-100)$
$-t = ln[\frac{1}{900}\cdot(\frac{100,000}{P}-100)]$
$t = -ln[\frac{1}{900}\cdot(\frac{100,000}{P}-100)]$
This function tells us how much time is required (in years) for a population to reach a level of $P$.
(c) We can find the time required for the population to reach 900:
$t = -ln[\frac{1}{900}\cdot(\frac{100,000}{P}-100)]$
$t = -ln[\frac{1}{900}\cdot(\frac{100,000}{900}-100)]$
$t = -ln(0.012345679)$
$t = 4.4$
It takes 4.4 years for the population to reach 900.
