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The population of Los Angeles
P(t) (in millions) can be approximated by the logistic growth function

P(t)=(3.194)/(1+14.589e^(-0.052 t))
where
t is the number of years since the year 1900 .
b. Use this function to approximate the population of Los Angeles on January 1, 2016.

The population of Los Angeles $P(t)$ (in millions) can be approximated by the logistic growth function $\newline$ $P(t)=\frac{3.194}{1+14.589 e^{-0.052 t}}$ $\newline$ where $t$ is the number of years since the year $1900$ . $\newline$ b. Use this function to approximate the population of Los Angeles on January $1$ , $2016$ .

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Convert between exponential and logarithmic form: all bases

Full solution

Q. The population of Los Angeles

P(t)

(in millions) can be approximated by the logistic growth function

\newline

P(t)=\frac{3.194}{1+14.589 e^{-0.052 t}}

\newline

where

t

is the number of years since the year

1900

.

\newline

b. Use this function to approximate the population of Los Angeles on January

1

,

2016

.

Calculate $P(t)$ : To find the population in $2016$ , we need to calculate $P(t)$ for $t = 2016 - 1900$ , which is $t = 116$ years.
Substitute $t$ into function: Substitute $t = 116$ into the logistic growth function $P(t) = \frac{3.194}{1 + 14.589e^{-0.052t}}$ .
Calculate exponent part: Calculate the exponent part: $-0.052 \times 116 = -6.032$ .
Calculate $e^{-6.032}$ : Calculate $e^{-6.032}$ . This is where we use a calculator because $e$ is an irrational number.
Plug into logistic function: $e^{-6.032} \approx 0.0024$ (rounded to four decimal places for simplicity).
Calculate denominator: Now, plug this value into the logistic growth function: $P(116) = \frac{3.194}{1 + 14.589 \times 0.0024}$ .
Divide to get population: Calculate the denominator: $1 + 14.589 \times 0.0024 \approx 1 + 0.0350 = 1.0350$ .
Divide to get population: Calculate the denominator: $1 + 14.589 \times 0.0024 \approx 1 + 0.0350 = 1.0350$ .Now, divide $3.194$ by $1.0350$ to get the population in millions: $P(116) \approx 3.194 / 1.0350$ .
Divide to get population: Calculate the denominator: $1 + 14.589 \times 0.0024 \approx 1 + 0.0350 = 1.0350$ .Now, divide $3.194$ by $1.0350$ to get the population in millions: $P(116) \approx 3.194 / 1.0350$ . $P(116) \approx 3.0865$ million.

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