Question

A city starts with a population of 500,000 people in 2007. Its population declines according to the equation

P(t) = 500,000e -0.099
where P is the population in t years later. Approximately when will the population be one-half the initial amount?

Answer 1

Answer: After 7 years the population will be one-half the initial amount.

Explanation:

Given: Initial population = 500,000

The population declines according to the equation:

`P(t) = 500,000e^( -0.099t)`, where P is the population in t years later.

One-half the initial amount = 0.5 x 500,000

= 250,000

Put P(t)=250,000, we get

`250000=500000e^(-0.099t)\\\\\Rightarrow\ (500000e^(-0.099t))/(500000)=(250000)/(500000)\\\\\Rightarrow\ e^(-0.099t)=\frac12\\\\\Rightarrow\ -0.099t=\ln \left((1)/(2)\right)\\\\\Rightarrow\ t=(1000\ln \left(2\right))/(99)=(1000(0.69314))/(99)\\\\\Rightarrow\ t=7.00148\approx7`

Hence, After 7 years the population will be one-half the initial amount.