The population, 𝑃, of Lake Norman with a large number of seasonal residents, can be modeled using the function 𝑃𝑡=3000 sin30𝑡+ 6100, where 𝑡 is the number of months after New Year’s Day. What is the minimum population?

Question

The population, 𝑃, of Lake Norman with a large number of seasonal residents, can be modeled using the function 𝑃𝑡=3000 sin30𝑡+ 6100, where 𝑡 is the number of months after New Year’s Day.   What is the minimum population?

supie · Answer

$\sf P(t)=10,000$$$P(t) = 5000 * sin(30(t - 7)) + 8000$$$$10,000 = 5,000 * sin(30(t - 7)) + 8,000$$~$$2,000 = 5,000 * sin(30(t - 7))$$~$\frac{2}{5} = sin(30(t - 7))$
$arcsin(\frac{2}{5}) = 30(t - 7)$
Where $$30(t - 7)=23.57818˚$$
So, $$(t - 7) =\frac {23.57818}{30}$$$$(t - 7) = 0.7859393$$$$t = 7.7859393$$