Question

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

Answer 1

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

Step-by-step 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

$250000=500000e^{-0.099t}\\\\\Rightarrow\ \frac{500000e^{-0.099t}}{500000}=\frac{250000}{500000}\\\\\Rightarrow\ e^{-0.099t}=\frac12\\\\\Rightarrow\ -0.099t=\ln \left(\frac{1}{2}\right)\\\\\Rightarrow\ t=\frac{1000\ln \left(2\right)}{99}=\frac{1000(0.69314)}{99}\\\\\Rightarrow\ t=7.00148\approx7$

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