A city starts with a population of 500,000 people in 2007. Its population declines according to the equation
P(t) = 500,000e -0.099t
where P is the population in t years later. Approximately when will the population be half the initial amount?
Answer:
2014
Step-by-step explanation:
Given the exponential growth function :
P(t) = 500,000e^-0.099t
Where P is population in t years later
When the population will 1/2 the initial amount
1/2 the initial amount = 1/2 * 500,000 = 250,000
P(t) = 250,000
250,000 = 500000e^-0.099t
250,000/500000 = e^-0.099t
In(0.5) = - 0.099t
t = −0.69314 / - 0.099
t = 7.0014
t = 7
Year which population = 500,000 = 2007
Hence, half the population will be attained on (2007 + 7) = 2014
After 7 years population of the city will be half of the initial amount.
Population function:
[tex]P(t) = 500000 e^{-0.099t}[/tex]
Initial population means population at t=0
P(0) = 500000
Half of initial population = 250000
So, we need to calculate the time t when the population will be half of the initial amount.
[tex]250000= 500000 e^{-0.099t}\\\\e^{-0.099t} = \frac{1}{2}[/tex]
Taking log both sides
-0.099t (log e)= log (1/2)
The logarithm of e is 1 when considering the base of the log as e.
So, -0.099 t= -0.693
t=7
Therefore, After 7 years population of the city will be half of the initial amount.
To get more about logarithm visit:
https://brainly.com/question/11495859
