Respuesta :
Answer:
The year is 1979.
Step-by-step explanation:
Given:
The function relating fish population and number of years since 1972 is:
[tex]f(x)=230(0.886)^x\\Where,\ f(x)\rightarrow \textrm{fish population}\\x\rightarrow \textrm{number of years passed after 1972}[/tex]
Now, when the fish population reaches 99 thousand, it means that [tex]f(x)=99[/tex]. Now, plugging in [tex]f(x)=99[/tex] in the above equation and solving for [tex]x[/tex]. This gives,
[tex]99=230(0.886)^x\\\frac{99}{230}=0.886^x\\\\\textrm{Taking natural log on both sides}\\\ln(0.886)^x=\ln(\frac{99}{230})\\\\\textrm{Using log property: }\log a^b=b\log a\\x\ln (0.886)=\ln(0.43)\\x\times (-0.12)=-0.84\\x=\frac{-0.84}{-0.12}=7\ years[/tex]
Therefore, the year after 7 years passed since 1972 is given as:
1972 + 7 = 1979.
So, the year when the number of fish reaches 99 thousand is 1979.
