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An initial population of 745 quail increases at an annual rate of 16%. Write an exponential function to model the quail population. What will the approximate population be after 4 years?
(2 points)
To write an exponential function to model the quail population, we can use the formula for exponential growth:
\[P(t) = P_0 * (1 + r)^t\]
Where:
- \(P(t)\) is the population after \(t\) years
- \(P_0\) is the initial population (745 quail)
- \(r\) is the annual growth rate (16% or 0.16)
- \(t\) is the time in years
Plugging in the values, we get:
\[P(t) = 745 * (1 + 0.16)^t\]
\[P(t) = 745 * 1.16^t\]
Therefore, the exponential function to model the quail population is \(P(t) = 745 * 1.16^t\).
To find the approximate population after 4 years, we can plug in \(t = 4\) into the function:
\[P(4) = 745 * 1.16^4\]
\[P(4) = 745 * 1.8607\]
\[P(4) \approx 1390.48\]
After 4 years, the approximate quail population will be 1390 quail.