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:
[tex]P(t) = 500,000e^{ -0.099t}[/tex], 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
[tex]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[/tex]
Hence, After 7 years the population will be one-half the initial amount.
